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(2006-02-21)   From Magmas to Semigroups to Monoids
In a monoid, the  associative  internal operator has a neutral element.

Bourbaki  calls  magma  a set endowed with  some  internal well-defined operation.  (Avoid the term  groupoid  for this, which is now best reserved for a concept in category theory.)

Multiplicative notations  are often used where the binary operator is understood between consecutive symbols representing elements.  Saying that an operation is  multiplicative  merely stresses the use of that convention  usually supplemented by the optional use of a so-called  multiplicative,  symbol,  (like a dot,  an x-shaped cross,  a  delta or any special  ad hoc  symbol)  whenever a clear separation between elements is deemed typographically appropriate.

Once a particular operator is so singled out, it's called a  multiplication  and the qualifier  multiplicative  can then be used,  especially to distinguish that from co-existing  additive  concepts.

Occasionally,  the multiplicative vocabulary is applied to several co-existing operators,  making distinct multiplicative symbols mandatory.  (Usage may or may not allow  one  such symbol to be dropped.)  For example,  the dot-product and the cross-product over  3D-vectors  are both construed as multiplications but the cross symbol can  never  be dropped.  In some international texts,  that cross is replaced by a  wedge,  which is the general symbol for an  exterior product,  of which the ordinary cross-product can be construed to be a special case.
 
When that particular operation is  not  denoted by a cross symbol,  it's called a  vectorial product  (Bourbaki:  produit vectoriel,  in French.)

Semigroups

If its operator is  associative,  a magma is called a  semigroup.  Associativity  is the property which makes the use of parentheses optional:

x y z   =   (x y) z   =   x (y z)

The  order  of a  finite  semigroup is its number of elements.  We count two semigroups as  distinct  when there's no  isomorphism  between them:

A semigroup operator may or may not be commutative.  (In a  commutative  semigroup,  xy  is the same as  yx  for any pair of elements  x  and  y.)

The numbers of distinct noncommutative semigroups are obtained from the above two tables,  by termwise subtractions.  Those numbers are always even because such semigroups come in pairs linked by an  anti-isomorphism.

Lone operators are often designed to be associative.  In complex structures with several operators, non-associativity may emerge in a natural way:

For example, in the realm of  hypercomplex numbers, the multiplication of  octonions  or  sedenions  is not associative.

Likewise, the cross-product in ordinary three-dimensional vector space is not associative.  Instead,  it verifies what's called  Jacobi's identity:

A × (B × C)   =   (A × B) × C  +  B × (A × C)

Incidentally,  any  vector space  endowed with an  anticommutative  internal operator  (×)  verifying that identity is called a  Lie algebra.

Monoids :

A  semigroup  in which there's a neutral element  e  is called a  monoid :

$e ,  "x ,     e x   =   x e   =   x

The number of noncommutative monoids is obtained by subtracting the corresponding entries of the above tables.  It's always an  even  number because such monoids come in pairs linked by an anti-isomorphism.

A monoid can also be defined as a category with a single object  (the arrows of that category being the elements of the monoid).

Magma   |   Quasigroup & Loop   |   Semigroup & Monoid
 
The semigroups of order 9 and their automorphism groups  by  Andreas Distler & Tom Kelsey  (2013-01-25).

(2006-03-04)   Invertible Elements in a Monoid
Two flavors of invertibility, which coincide when both exist.

In a monoid, an element  x  is said to be  right invertible  if there's a  right-inverse  x'  of  x  (which is to say that the product  xx'  is unity).  It's called   left invertible  if there's a  left-inverse  x''  (such that  x''x  is unity).

When both inverses exist, they are necessarily equal  (HINT:  Consider  x''xx' ). In this case,  x  is said to be  invertible  and its (unique) inverse is denoted  x^-1.

The Group  M*  of the Invertible Elements of  M :

In a multiplicative monoid  M, the set of all invertible elements form a group which is often denoted  M*.  (The neutral element of  M  belongs to  M*.)

For example :

• *  is the multiplicative group of nonzero real numbers. 
• *  is the multiplicative group of nonzero complex numbers. 
• (/n)*  is the finite multiplicative group consisting of the  f(n)  residues modulo n which are coprime with n  (f being Euler's totient function).  That's the structure underlying Dirichlet's character tables.

Cancellativity (cancellable element  & cancellative operation)

(2006-02-21)   The Free Monoid  (strings over a given alphabet)
A particular monoid where  only  the neutral element is invertible.

All the finite strings (or words) whose characters (letters or symbols) are taken from a given  alphabet  form a monoid under the operation of  concatenation  (concatenating two strings means appending the second to [the right of] the first).  The concatenation of two strings called  A  and  B  is best called  "A before B".

The  empty string  is the neutral element for  concatenation.

This monoid is  free  from any relations equating  distinct  strings of basic symbols.  Hence the name  (French:  monoïde libre ).

Clearly, concatenating two nonempty strings yields something other than the empty string.  The empty string is thus the only string with an inverse...

The free monoid over an alphabet of only one symbol is isomorphic to the natural integers endowed with addition  (0,1,2,3...).  In every other case, a free monoid is clearly  not  commutative.

Free Object

(2013-01-02)   Exponentiation   (for power-associative "multiplications")
Raising something to the power of an integer.

Whenever some kind of associative  multiplication  is defined,  something like  x³  is simply shorthand for  xxx.  It's always legitimate to raise an element to the power of a  positive  integer that way.

The n-th power of an element can be well-defined even for non-associative multiplications.  The weaker property of alternativity suffices  (e.g., the multiplication of  octonions  is just alternative).
 
By definition, the  weakest  property of multiplication which allows a well-defined exponentiation  (to the power of a positive integer)  is called  power associativity.  Almost all "multiplications" have it.

x⁰  is always defined as equal to the  neutral element,  if there is one  (otherwise, it's undefined).  It makes no difference whether  x  is  invertible  or not  (with ordinary arithmetic, zero to the power of zero equals  one).

One enlightening example is that of the  free monoids, defined in the previous section:  xⁿ  denotes the concatenation of  n  strings identical to  x.  Thus,  x⁰  denotes the empty string  (the concatenation of no strings).  x  need not be "invertible".

If  x  is invertible,  x^-3  simply denotes  (x^-1)³.  Only invertible elements can be raised to the power of a  negative  integer.

Empty sums, empty products, empty intersections, etc.

Raising something to the power of zero is a special case of an  empty product.  The result of not performing at all some well-defined associative operation depends on that operation alone:  It's equal to its neutral element  (whenever it has one).

• An empty sum is  0  (the neutral element for addition).
• An empty product is  1  (the neutral element for multiplication).
• An empty categorial product is the terminal element  (if there's one).
• An empty union of subsets is the empty set.
• An empty intersection of subsets is the whole set.
• An empty concatenation of  strings  is the empty string.
• The  lowest common multiple  of no positive integers is  1.
• The  greatest common divisor  of no nonnegative integers is  0.
• Etc.

Empty product on the HP Prime calculator

(2006-02-21)   Group
A group is a  monoid  in which  every  element is invertible.

Walther von Dyck (1856-1934) formally defined  groups  in 1882:

A group is a set  G  on which an internal operation is defined which verifies the following properties  (using multiplicative notations for the operator).

• Closure :  "xÎG,  "yÎG,   x y  Î G   (The product is  well-defined.)
• Associativity :   "xÎG,  "yÎG,  "zÎG,   (x y) z  =  x (y z)
• A  unity element  (e)  exists :   $eÎG,  "xÎG,   e x  =  x e  =  x
• Universal Invertibility :   "xÎG,  $x'ÎG,   x x'  =  x' x  =  e

G  is called a  commutative group  (or  abelian  group)  when we also have:

• Commutativity (optional) :   "xÎG,  "yÎG,   x y  =  y x

Niels Abel (1802-1829)  considered algebraic equations whose Galois groups are commutative  (and went on to prove that they are solvable).  Leopold Kronecker (1823-1891)  called such equations Abelian and the qualifier was first applied to all commutative groups by  Camille Jordan (1838-1922)  in 1870.  The term is now so common in that capacity that it's usually lowercased  (abelian).

An  additive  group is merely a group  (usually abelian)  where additive notations are used:  The  plus  sign  (+)  denotes the group operator.

Additive notations are rarely used for a  noncommutative  operator.  (In a  ring,  addition is  always  commutative.  In a  near-ring,  it doesn't have to be.)  One well-known exception is the addition of transfinite ordinals  à la  Cantor  (which I dare consider a misguided effort).

Single-sided  group  properties imply double-sided ones :

The double-sidedness of two of the above  group axioms  need not be postulated; it can be derived from one-sided equivalents of those axioms :

• There's a  right-neutral  element  e  :    "x,   x e  =  x
• Every element is  right-invertible :   "x, $x',   x x'  =  e

Indeed, we may compute   x' x   using just those two single-sided postulates:

x' x   =   x' x e   =   x' x x' (x' )'   =   x' e (x' )'   =   x' (x' )'   =   e

That would show  x'  to be the inverse of  x,  if we knew that  e  is neutral  on both sides.  That fact is easy to prove,  using the above as a lemma:

"xÎG,   e x   =   (x x' ) x   =   x (x' x)   =   x e   =   x

---

This double-sided neutrality implies that there's only one unity  e .  (HINT:  Assume another unity  e'  and consider  e e' ).

Similarly, there's  only one  inverse  x'  of  x  (HINT:  Let  x"  be another and consider x' x x" ).  So we may safely talk about  the  inverse of x.

Note, finally, that   (x' )' = x   (HINT:  x' (x' )' = e ).

(2006-02-21)   Subgroups
A subgroup is a group contained in another group.

A subgroup H of a group G is a subset H of G which forms a group under the group operation defined over G.  H is a subgroup of G  if and only if  it contains the product of any element of  H  by the  inverse  of any other element of  H.  A multiplicative subgroup is said to be  stable by division.

"xÎH,  "yÎH,    x y^-1 Î H

When  additive  nomenclature and notations are used, this translates into the following statement, which says that a subgroup of an additive group is merely a subset that's  stable by subtraction :

"xÎH,  "yÎH,    x - y Î H

A  proper  subgroup of G is a subgroup of G not equal to G itself.  The  trivial  group  {e}  has no proper subgroup.

Any  intersection  of subgroups is a subgroup.

The  centralizer  in a group  G  of a subset  E  consists of all the elements of  G  which commute with every element of  E.  It is a subgroup of G.

The centralizer in  G  of  G  itself is the center of  G,  denoted  Z(G)  (it's the intersection of all centralizers in  G).  The center is a normal subgroup of G, but other centralizers may not be.  Elements of  Z(G)  are called  central.

(2020-01-27)   Ideals of a multiplicative  semigroup  A
A multiplicatively absorbent subset of  A  is an  ideal  of  A.

By definition:
For a  left-ideal  I, the product  ax  is in I whenever x is:   "aÎA, aI Í I
For a  right-ideal  I, the product  xa  is in I whenever x is:   "aÎA, Ia Í I
Unless otherwise specified, an  ideal  is  both  a right-ideal and a left-ideal.  Note that the empty set is an ideal of any semigroup.

This is most often used in the context of  ring theory,  where an ideal of a ring  (not necessarily a unital  ring)  is defined as being both a  multiplicative  ideal  (in the above sense)  and an  additive  subgroup  (thus, the empty set is not an ideal of a ring because it's not a group).

Any intersection of [one-sided] ideals is a [one-sided] ideal.  The intersection of all the ideals of a semigroup is called its  minimal  ideal.  If it's nonempty,  the minimal ideal  M  of a  commutative  semigroup is a group.  This is to say that  M  has a neutral element, even if the whole semigroup doesn't.

(2006-03-09)   Generators of a Group
The smallest  subgroup  containing  E  is said to be  generated  by  E.

For any subset  E  of a group  G,  the intersection of all  subgroups  of  G  containing  E  is a subgroup of  G,  called the subgroup  generated  by  E.

E  is said to be a set of  generators  of whatever subgroup it generates.  A group which is generated by a finite set is said to be  finitely generated.

For example, the  additive  group  (,+)  of the integers is generated by the set {1}.  It's also generated by {2,3} or any other pair of  coprime  integers  (because of  Bezout's lemma).  More generally,  (,+)  is generated by any set of coprime integers  (not necessarily  pairwise  coprime)  like  {6,10,15}.

A  finite  group  (of order  n )  which is generated by a single element is a  cyclic  group.  An element of such a group which generates the whole group is called a  primitive  element  (or a primitive root,  with the vocabulary inherited from representing the cyclic group of order  n  as the "n-th roots of unity" in  complex numbers).  There are  f(n)  different elements in a cyclic group which are primitive ones  ( f  being Euler's totient function).

The multiplicative group  (⁺, ´)  of positive rationals is  not  finitely generated.  It's generated by the  prime numbers  {2,3,5,7,11,13,17,19...}.

Additive  groups which are  not  finitely generated include the rationals,  the reals,  the  complex numbers,  the  p-adic integers,  the p-adic numbers,  etc.

(2016-05-29)   Presentations of a group:  Generators and  relators.
Describing a group using the relations obeyed by its  generators.

A finitely-generated group can be described by naming a set of generators and stating the nontrivial relations they obey  (the  relators).  Those relators are normally given by expressions which are equal to the neutral element  (minimally so)  but explicit equations are also commonly used.

A  free group  has no relators.  The simplest free group is isomorphic to the additive group of the integers  (,+)  and has the following multiplicative  presentation,  which names a single generator and states no relators:

< a | >

Less trivially,  the  octic group  D₄  could be  presented  as follows:

<  r, s  |  r ⁴, s ², srsr  >         or         <  r, s  |  r ⁴ = s ² = srsr = 1  >

Do not confuse such  presentations  with  (linear)  representations.

(2006-03-02)   Cosets, Index and Lagrange's Theorem
The order of a subgroup divides the order of the group.

By definition, the  order  |G|  of a finite group  G  is its number of elements.  (The order of an element x is the order of the subgroup generated by {x}.)

Cosets :

In a group  G,  the  left-coset  of an element  x,  with respect to the subgroup  H,  is the subset  x H  of  G  (consisting of all products   x h   where  h  is an element of  H).  Similarly, the  right-coset  is  H x.

Index of a Subgroup :

Two left-cosets with respect to H are either disjoint or identical and they have the same  cardinality  as H  (i.e., the same number of elements  if  finite).  Whenever it's finite, the number of left-cosets with respect to H is equal to the number of right-cosets.  It's denoted  [G:H]  and is called the  index  of  H  in  G.

Lagrange's Theorem :

In the case of a  finite  group  G,  the fact that such left-cosets form a  partition  of  G  shows that the order of the subgroup  H  divides evenly the order of  G.

This result is known as  Lagrange's Theorem.  It's now presented as one of the most basic results of  Group Theory,  named in honor of Joseph-Louis Lagrange (1736-1813), who made a related remark in 1777.  The general result was probably known to Cauchy (1789-1857) but it was only formally proved in 1861,  by  Camille Jordan (1838-1922; X1855).

Commensurability :

Two subgroups are said to be commensurable when the index of their intersection is finite in each of them.  The qualifier is inherited from ancient Greek mathematics, where two real numbers are called commensurable when they are proportional to two integers.  The two additive groups generated by two such numbers are indeed commensurable in the above sense (their intersection is the additive group generated by the  lowest common multiple  of the two numbers).

Lagrange's theorem   |   Cosets and Lagrange's Theorem (9:18)  by  Liliana De Castro  (Socratica, 2017-03-20).

(2020-05-19)   Cauchy's group theorem   (Cauchy, 1845)
If a prime number  p  divides |G|,  then some element of G has  order  p.

The commutative case can be used as a lemma to prove Cauchy's theorem and also its generalization by Sylow  (Sylow's first theorem, 1872).

Lemma :   Cauchy's group theorem  holds for abelian groups.

Proof :   Let G be an abelian group G whose order is a multiple of the prime p:   |G| = n = m×p.  First,  we see that the proposition is true if G is cyclic,  generated by element  a  (since the element  a^m  is then of order p).

Otherwise, we proceed by  induction  on  m,  starting with the case  m = 1  which makes p the order of G.  This is trivial because,  by  Lagrange's theorem,  the order of an element must divide the order of the group and can thus  only be 1 or p.  In other words, any element of G besides identity is a satisfactory element of order p  (which establishes also that G is cyclic).

For  m ≥ 2,  consider an element  h  of G, besides identity.  Let H be the nontrivial subgroup  generated  by h.  H is a normal subgroup  (as is any subgroup in the abelian case).  Both H and G/H are nontrivial group of order strictly less than n  (because we've already disposed of the cases where G is cyclic).  Since the product of their orders  (respectively |G| and [G:H])  equals n = m×p,  at least one of them is divisible by p.  In either case,  the induction hypothesis implies that the corresponding group contains an element of order p.  Either way, we can use that to obtain an element of order p in G, as follows:

• If H contains an element x of order p,  then x is also in G and we're done.
   
• If G/H contains an element y of order p,  then y is the class xH of some element x of G.  For any integer k, the class of x^k is y^k.  So, the order of x is the same as the order of y, namely p.  

This concludes the proof that Cauchy's theorem holds for abelian groups.

Proof for non-abelian groups :

Cauchy's group theorem (1845)   |   Augustin-Louis Cauchy (1789-1857)
 
Cauchy's Theorem in Group Theory  by  Robin Whitty  (Theorem of the Day #227).

(2020-05-19)   Sylow's Theorems  [pronounced see-lov ]   (1872)
On the number of subgroups of given order in a finite group  G, 

For a  prime  p,  a  p-group  is a group where the order of any element is a power of  p.  If it's a  subgroup  of G, it's called a  p-subgroup  of G.

In a finite group  G,  a Sylow p-subgroup  (abbreviated p-SSG)  is a  maximal  p-subgroup of G.  The set of all p-SSG is denoted  Syl_p (G).  Remarkably, all of those are isomorphic to each other.

Sylow's First Theorem :

Sylow's Second Theorem :

Sylow's Third Theorem :

p-group   |   Sylow theorems (1872)   |   Peter Ludwig Sylow (1832-1918)
 
Groups of orders 56 are not simple (17:05)  by  Suraj Pr.  (Mr. SRJ, 2018-09-08).
 
Visual Group Theory: The Sylow theorems (48:36)  by  Matthew Macauley  (2016-04-05).
 
Sylow's theorems  by  Robin Whitty  (Theorem of the Day #258).

(2006-03-02)   Normal Subgroups  and  Quotient Groups
The left and right cosets with respect to a normal subgroup are identical.

The concept of a  normal subgroup  is due to  Evariste Galois  (1832).

A subgroup  H  is  normal  when  aH = Ha  for any a.  Such a subgroup is also called  invariant  or  distinguished  (French:  sous-groupe distingué ).

A subgroup  H  is normal  iff  it's stable under  any  inner isomorphism.

"aÎG, "xÎH,    a x a^-1 Î H

Quotient Group of a  Normal  Subgroup :

To a  normal  subgroup H  corresponds an  equivalence relation  among elements of  G  defined by calling x and y equivalent when  xy^-1  is in  H  (in other words, when x and y have the same  left cosets with respect to H).

The  equivalence classes  so defined form a group denoted  G/H  and called the  quotient  of  H  in  G  (or of  G  by  H)  also dubbed  "G modulo H".

Although the above  equivalence relation  is defined for any subgroup  H,  the equivalence classes form a group  only  when  H  is normal.

Examples of Normal Subgroups :

Any group  G  is a  normal subgroup  of itself  (the only  non-proper  one).

The  trivial group  {e}  is a  normal subgroup  of any group  G  whose neutral element is  e.  (It's a  proper  subgroup of any such  G  but itself.)

The derived subgroup  G'  is also always a normal subgroup of  G.

The  center  Z(G)  of a group  G  consists of all the elements which commute with  every  element  G.  A member of  Z(G)  is called a  central element.  A  noncentral  element is an element which doesn't commute with at least one other element.  The  center  is a normal subgroup.  So is any subgroup of the  center  (in particular, any subgroup of an  abelian group  is normal).

If  f  is a homomorphism or an antihomomorphism from group  G,  then the kernel of  f  (ker f )  is a normal subgroup of  G.  More generally, so is the inverse image  (pre-image)  of any normal subgroup of  f (G).  For a normal subgroup  H  of  G,  the direct image  f (H)  is a normal subgroup of  f (G).

For any subset E of the group G, the  subgroup generated by all the conjugates of the elements of  E  is called  conjugate closure  of E.  It's a normal subgroup containing E.  In fact, it's the smallest normal subgroup containing E  (i.e, it's the intersection of all normal subgroups containing E).  It's thus also known as the  normal closure  of E.

Any Subgroup is a Normal Subgroup of its Normalizer :

The  normalizer  of a subgroup  H  consists of all elements  x  of the group  G  for which  x H = H x  (in particular all elements of  H  belong to its normalizer).  The normalizer of  H  is a subgroup of  G.  By definition,  H  is a  normal subgroup  of its normalizer  (H  need not be a  normal subgroup  of the whole group G).

(2019-04-13)   Wielandt Symbols   (Helmut Wielandt, c. 1960)
Notations for  [proper]  normal subgroups  (or  ideals).

Two conventions are floating around to distinguish between a standard (reflexive)  ordering relation  and its  strict  (antireflexive) counterpart:

The above notations for  normal subgroups  were introduced by  Helmut Wielandt  around 1960.  They are now also used to denote  ideals  in  ring theory  (since an ideal is to a ring what a normal subgroup is to a group).

Helmut Wielandt (1910-2001)
Notation for proper normal subgroup  (or proper ideal)   TeX - LaTeX  Stack Exchange

(2006-04-05)   Homomorphisms and Anti-homomorphisms
Functions for which the image of a product is a product of the images.

An  homomorphism  is a  map  (or function)  which preserves some specific algebraic operation(s).  A  group homomorphism  is thus a map  f  from a [multiplicative] group  G  into another group  H,  which is such that:

"xÎG,  "yÎG,     f (x y)  =  f (x) f (y)

If  f  is surjective  ("onto" H)  it's called an  epihomomorphism  (or "homomorphism onto").  If it's injective  ("one-to-one")  it's called an  monomorphism. If it's bijective  ("one-to-one onto")  it's an  isomorphism.

An homomorphism from G  to itself is called an  endomorphism  of G.  A bijective endomorphism is called an  automorphism.

The automorphisms of a group  G  form a group, denoted  Aut(G).

Anti-homomorphisms :

An  anti-homomorphism,  with respect to a multiplicative operator, is a function  f  which reverses the order of that multiplication :

"xÎG,  "yÎG,     f (x y)  =  f (y) f (x)

In any group,  inversion  is an example of an anti-homomorphism:

( x y ) ^-1   =   y ^-1  x ^-1

The concepts defined above for homomorphisms have their counterparts for anti-homomorphisms:  Anti-epihomomorphism,  anti-monomomorphism,  anti-isomorphism,  anti-endomorphism  and  anti-automorphism.

Kernel   (French:  noyau )

For a homomorphism  (or an anti-homomorphism)  f  from group  G  to a group of identity  e,  the  kernel  of  f  is a normal subgroup of  G  defined by

ker  f   =   { xÎG  |  f (x) = e }

(The homomorphic pre-image of  any  normal subgroup is normal.)

Kernel of a ring homomorphism   |   Categorial kernel
 
Kernel of a Group Homomorphism (4:52)  by  Liliana De Castro  (Socratica, 2016-05-02).

(2006-03-05)   Sym(E)  =  Symmetric Group on  E   (Gersonides, 1321)
The group of the permutations of  E  (bijections of the set E onto itself).

A  permutation  of  E  is a one-to-one correspondence (bijection) of  E  onto itself.  The term is most commonly used when  E  is finite, but it's also acceptable when  E  is infinite  (possibly uncountably so).

The permutations of  E  are a group under  function composition  (o).

f o g (x)   =   f ( g (x) )

In the finite case, the  symmetric group of degree n  is denoted  S_n.  Its order is the number of permutations of  n  elements, namely  n!  ("n factorial").

The largest  order  of an element in the symmetric group  S_n  is traditionally denoted  g(n)  where  g  is called  Landau's function,  in honor of the German mathematician  Edmund Landau (1877-1938)  who proved the following  asymptotic  equivalence in 1902:

Log g(n)   ~   (n Log n)^½

Even  permutations form the  alternating group  A_n  (whose order is  n!/2 ).  It's the derived subgroup of the symmetric group:  A_n = S'_n

An even permutation is obtained by an even number of switches (swaps of two elements).  The parity, or signature, of a finite permutation may be determined by counting its number of inversions.

Notations for small permutations :

One standard way to record computations in the realm of  very small  finite groups is to use a string of different characters  (digits or letters)  to denote the permutation which transforms the sorted elements in the top row into the matching elements of the bottom row.  Both row are placed between parentheses.  Juxtaposition of two such notation indicates the  composition  of the functions so denoted,  with the usual convention that the rightmost function is to be applied first  (composition isn't commutative):

Cycle decomposition of a permutations :

A  cyclic permutation  of  n  elements is denoted by a sequence between parentheses.  The image of an element is the element to its right  (the last element is mapped back to the first one).  There are  n  equivalent ways to denote such a permutation,  since there's a free choice of which element is written first in the list.  Equivalent notations are equated:

(1 4 5 3 2)   =   (4 5 3 2 1)

A  cycle  is a permutation of  n  elements which is a  cyclic permutation  of  m  of those elements  (m ≤ n)  which leaves the others unchanged.  Two or more cycles are said to be  disjoint  when operate on different elements  (each cycle applies only to elements which are left unchanged by the others).  Any permutation can be decomposed as a composition of disjoint cycles in a unique way  (up to the order of those cycles, which is irrelevant since disjoint cycles commute).  Our  previous example  entail cycles which do not commute because they're  not  disjoints.  namely:

(3 4) (2 3)   =   (2 4 3)                   (2 3) (3 4)   =   (2 3 4)

A cycle of order 2  (a 2-cycle)  is called a  switch  or a  transposition.  It's useful to know that the  signature  of a cycle of order  n  is  (-1)ⁿ⁺¹.

Cycle Notation for Permutations (12:36)  by  Liliana De Castro  (Socratica, 2018-10-15).

Cayley's Group Theorem  (1878) :

Arthur Cayley (1821-1895) observed that a group  G  is always isomorphic to a subgroup of  Sym(G).

Proof :   In the multiplicative group  G,  we associate to an element  a  the bijection  T(a)  which sends an element  x  to  ax .  T is an  injective  homomorphism  (i.e.,  a  monomorphism)  from  G  to  Sym(G),  which is called the  regular representation  of G.

T(a) o T(b)   =   T(a b)

So,  any  finite group of order  n  is isomorphic to a subgroup of  S_n .  

Numericana :   Yoneda lemma  (Category theory)

(2006-03-02)   Inn(G):  The Group of Inner Automorphisms on  G
An inner automorphism is a conjugation by a given element of  G.

To any element  a  of  G  is associated a special type of automorphism, called an  inner  automorphism  (French:  automorphisme intérieur )  defined as follows  ( f_a  is called  conjugation by a ).

" x,   f_a(x)  =  a x a^-1       [ Note that  f_a o f_b  =  f_ab ]

Under function composition,  inner automorphisms  form a normal subgroup  (see proof later in this section)  denoted  Inn(G), of the group of the automorphisms on G, denoted  Aut(G)  (itself a subgroup of Sym(G), the symmetric group on G).  Conjugation by  a  is the identity function just if  a belongs to the center of  G.  Consequently:

Inn(G)  is isomorphic to the quotient of  G  by its center.

Note that a subgroup  H  of  G  which is mapped onto itself by  any  inner automorphism is a  normal subgroup  (also called  invariant subgroup).

More generally,  two subgroups of G are said to be  conjugates  of each other when there is an  inner isomorphism  between them.

---

The above claim that  Inn(G)  is a  normal subgroup  of  Aut(G)  is established by showing that conjugation by  any  automorphism  g  of an inner automorphism  (conjugation by  a)  yields another inner automorphism.  That can be proved in a single line:

" x,     g o f_a o g^-1 (x)   =   g ( a g^-1(x) a^-1 )   =   g(a) x g(a)^-1   =   f_g(a) (x)

(2006-03-02)   Out(G):  The Outer Automorphism Group
The members of  Out(G)  are  classes  of automorphisms of G.

The  outer automorphism group  of the group  G  is defined as the  quotient  of its  group of automorphisms  by its group of  inner automorphisms :

Out(G)  =  Aut(G) / Inn(G)

Unfortunately, the elements of Out(G) are known as outer automorphisms although they're not "automorphisms" at all !

(2014-12-17)   Complete Groups
A concept  unrelated  to the completeness of metric or uniform spaces.

A group  G  is said to be  centerless  when its center is trivial,  which is to say that only the identity element commutes with every element.

A  complete group  is a centerless group whose only automorphisms are  the inner  ones.  (Equivalently, it's a group whose center and  outer automorphism group  are trivial.)

If a group  G  is complete, it's isomorphic to  Aut(G)  (its automorphisms).  However, the converse need not be true  (one counterexample is D₄ ).

"Classification of small complete groups" (2010) in   Math Exchange   and   Math Overflow.

(2020-09-14)   The orbit-counting lemma
Variously named after Cauchy (1845), Frobenius (1887) or Burnside...

Orbit-counting theorem (Cauchy-Frobenius lemma)   |   Redfield-Pólya theorem (1927,1937)

(2006-03-20)   Conjugates and the  Conjugacy Class Formula
The conjugacy classes of a group G form a partition of G.

Two elements  x  and  y  of a group  G  are said to be  conjugates  when there's an inner automorphism from one to the other, that is, when there's an element  a  of  G  such that  ax = ya.

So defined,  conjugacy  is an  equivalence relation  (it's reflexive, symmetric and transitive). The  conjugacy class  of an element  x  is the set of all elements of  G  which are conjugate to it.  Every element is in one and only one of those classes  (equivalence classes always form such a  partition). 

If  x  is in the center of G,  denoted Z(G), then the conjugacy class of  x  is  simply  {x}  (a set of only one element).  More generally, we would establish that the number of elements that are conjugate to  x  is equal to the index in  G  of the centralizer  C  of  {x}.  That number is usually denoted  [ G : C ].

Tallying the conjugacy classes with more than one element by assigning each a different index  i,  we obtain the so-called  conjugacy class formula :

| G |   =   | Z(G) |  +  å _i  [ G : C_i ]

The second term is an empty sum  (equal to zero)  when G is commutative.

(2006-03-05)   Simple Groups
A group is  simple  when it has just  two  normal subgroups.

{e}  and  G  are trivially always  normal subgroups  of  G.  The group  G  is said to be  simple  when its only  normal subgroups  are those  two.

Just like 1 isn't said to be prime, the trivial group  {e}  isn't called "simple".

Simple groups (8:52)  by  Liliana De Castro  (Socratica, 2018-01-10).

(2006-03-06)   Derived Subgroup  G'  (or "Commutator Subgroup")
G',  G⁽¹⁾  or [G,G]  is the subgroup of G generated by its commutators.

The commutator  [x,y]  of two elements of the multiplicative group  G  is:

[x,y]   =   x y x^-1 y^-1   =   x y (y x)^-1

The set of all commutators isn't necessarily a subgroup.  What's called the  derived subgroup  (or commutator subgroup)  is the subgroup they generate  (i.e., the smallest subgroup which includes all commutators).

A group is said to be  perfect  when it's equal to its derived subgroup.  In particular,  a group which contains only commutators is perfect.  That's so for all  finite non-abelian simple groups,  as was first conjectured by  Øystein Ore (1899-1968)  in 1951.  Ore's conjecture  was  proved  in 2010,  using the  classification  of finite simple groups.

The derived subgroup of a group is a normal subgroup,  as the following identity demonstrates  (since the set of commutators is thus shown to be stable under  any  inner automorphism, so is the subgroup they generate).

a [x,y] a^-1   =   [ axa^-1, aya^-1 ]

G'  is also the  smallest  normal subgroup of  G  whose  quotient group  in  G  is  abelian  (i.e., commutative).  The group  G/G'  is known as the  abelianization  of  G  (it's the largest abelian quotient in G).

Examples of Derived Subgroups :

The derived subgroup of any  abelian group  is the  trivial  subgroup.

The derived subgroup of the symmetric group  S_n  is the  alternating group  A_n.  The derived subgroup of the alternating group is itself:  A'_n = A_n.

The derived subgroup of the Quaternion group  is  {+1,-1}.

Derived Series :

It's the sequence where the  n+1 st  term is the  derived subgroup of the n-th one  (starting with the whole group when  n = 0).

Commutator Subgroup   |   Commutator   |   Derived series   |   Perfect core

(2006-03-21)   Direct Product  (or Direct Sum)
The group made from the independent juxtaposition of several groups.

The  direct product  of two groups  G  and  H  is the group obtained by using for the cartesian product  G ´ H  independent operations on the components:

(g,h) (g',h')   =   ( g h , g'h' )

The term  direct sum  is used for the same concept with additive notations:

(g,h) + (g',h')   =   ( g+h , g'+h' )

Similar rules can be used for cartesian products of any number of monoids.

Extensions to infinitely many components :

The concept extends naturally to  direct sums  (or  direct products )  of infinitely many monoids.  Such direct sums are usually understood to be  finitely restricted  (by considering just the elements having only a finite number of components that differ from the relevant neutral element).

This assumption is always made in the case of  vector spaces  (only finitely many components are nonzero in the resulting structure)  and it's prudent to clearly distinguish between the two possibilities for infinite cartesian products endowed with component-wise operations.

For example, the fundamental theorem of arithmetic provides a standard isomorphism between the multiplicative monoid of the positive integers and the finitely restricted direct sum of infinitely many copies of the nonnegative integers  (each such copy being associated with a prime number).  Using standard notations, this can be expressed as:

( * , ´ )   =   (

()

, + )

Note that the set appearing in the right-hand-side of the above is countable, because of the parenthesized exponent which indicates a finite restriction in the above sense.  A lack of parentheses around the exponent would denote an uncountable set which is rarely investigated, if ever  (that beast includes elements idenfified with products of infinitely many coprime integers).

(2016-01-10)   Fundamental Theorem for Finite Abelian Groups
Finite abelian groups are either cyclic or  direct sums.

Thus, if  n  is the  k-th  power of a prime, the number of  non-isomorphic  abelian groups is equal to the number  p(k)  of  partitions  of k.

More generally, if the  prime factorization  of  n  is   q₁^k1 q₂^k2... q_m^km   then the number of non-isomorphic abelian groups of order  n  is equal to:

Abel ( n )   =   p( k₁ )  p( k₂ )  ...  p( k_m )

That's a  multiplicative function  of  n.  which depends only on its  prime signature  (A000688).

For what n are there one million abelian groups of order n ?

By trying only the first 61,  we see that the only partition numbers which divide  1000000  are   p(1) = 1,  p(2) = 2  and  p(4) = 5.  Therefore, there are  exactly  1000000 distinct abelian groups of order  n  if and only if the factorization of  n  consists of:

• 6  primes of multiplicitity  4.
• 6  primes of multiplicitity  2.
• Any number of primes of multiplicitity  1  (possibly none).

The smallest example is a  35-digit  integer:

(2 . 3 . 5 . 7 . 11 . 13)⁴  (17 . 19 . 23 . 29 . 31 . 37)²   =   4.96597898...10³⁴

Exactly one million abelian groups of order n?   (Mathematics Stack Exchange, 2014-05-08).

(2019-04-11)   Solvable Groups

Solvable group

(2019-04-11)   Burnside Theorem

Burnside theorem (1904)   |   William Burnside (1852-1927)

(2014-12-21)   Hol(G) :  Holomorph group of the group G.
A semi-direct product of  G  and its  group of automorphisms  Aut(G).

If  f  is a  homomorphism  from a group  H  to  Aut(G),  the  semi-direct product  of  G  and  H  with respect to  f  is the group denoted  G ´_f H  consisting of the cartesian product  G ´ H  with the  multiplication :

(x,a)  (y,b)   =   ( x f (a) (y) ,  ab )

When  f  is the  trivial homomorphism  (i.e.,  f (a)  is the identity of  G  for any a)  this  semi-direct product  is just the  direct product  of  G  and  H.

Holomorph :

When  H  is equal to  Aut(G)  we may use the identity of  Aut(G)  as the homorphism  f  appearing in the above definition and define the  holomorph  Hol(G)  as the  semi-direct product  of  G  and  Aut(G)  in which :

(x,a)  (y,b)   =   ( x a(y) ,  ab )

Holomorph of a group   |   Semidirect products (inner or outer)
 
The Automorphisms of the Automorph of a Finite Abelian Group (1956)  by  William H. Mills  (1921-2007?).

(2006-03-05)   Some Finite Groups
Groups of small orders and their families...

Additive notations  (using the symbol "+" for the operator)  are often used for commutative groups  (abelian groups).  Groups isomorphic to the group  C_n = (/n, +)  of residues modulo n are called  cyclic groups.

The above is a special case of the notation  A/I  which denotes a ring obtained as a  quotient of a ring by one of its ideals.  As such it's a structure endowed with  two  operators.  The single-operator additive group of that structure is properly denoted  (A/I,+).  The expression  (A/I,+,x)  is pleonastic.  The notation  Z_p  is for  something else.

Cyclic Group  C₅

+	0	1	2	3	4
0	0	1	2	3	4
1	1	2	3	4	0
2	2	3	4	0	1
3	3	4	0	1	2
4	4	0	1	2	3

All groups of  prime  order are  cyclic  (as Lagrange's Theorem implies that the subgroup generated by a nonneutral element is equal to the entire group).  The same is true for groups whose order is a  cyclic number  (i.e., an integer coprime to its Euler totient).  That result is due to William Burnside.

The smallest  noncyclic  groups are thus of order  4 and 6.  The  Klein group  is the noncyclic group of order 4.  The smallest  noncommutative  group is the following group  S₃ = D₃  (the 6 symmetries of an equilateral triangle).

Klein Group

+	0	1	2	3
0	0	1	2	3
1	1	0	3	2
2	2	3	0	1
3	3	2	1	0

Dihedral Group  D₃

	A	B	C	D	E	F
A	A	B	C	D	E	F
B	B	C	A	E	F	D
C	C	A	B	F	D	E
D	D	F	E	A	C	B
E	E	D	F	B	A	C
F	F	E	D	C	B	A

The  Klein Group  (V)  is isomorphic to the  direct sum   C2 ´ C2 Felix Klein  called it  Vierergruppe  in 1884.

 

The  dihedral group  D_n consists of the 2n symmetries of a regular n-gon  (n rotations, n flips).

When he proposed the  cyclic structure of benzene  in 1865,  August Kekulé (1829-1896)  thought that the  C₆H₆  molecule had  trigonal symmetry  (expressed by the order-6 group  D₃  tabulated above)  because of his vision that single and double bonds were alternating along the carbon ring.  The currently accepted symmetry for the benzene molecule is the hexagonal group  D₆  (of order 12)  with 3 of the binding electrons in a delocalized orbital covering the whole ring.

There are 5 groups of order 8.  Three are  abelian :  C₈  and the two direct sums  C₂+C₄   and  C₂+C₂+C₂  (the additive group of the field of order 8).  The other two groups of order 8 are  noncommutative,  namely the dihedral group D₄  (the symmetries of a square) and the  quaternion group  Q₈ :

(2006-03-05)   Quaternion Group  Q₈  &  Quaternions   (Hamilton, 1843)

On October 16, 1843, the fundamental equations below  (which imply the given multiplication table)  occurred at once to Hamilton as he was crossing   Brougham Bridge    (Broom Bridge)  in Dublin.  He carved them into the stone of the bridge  (the original carving is gone but a plaque celebrates this act of "mathematical vandalism").

i 2   =   j 2   =   k 2   =   i j k   =   -1		Quaternion Group   Q8   1 i j k -1 -i -j -k 1 1 i j k -1 -i -j -k i i -1 k -j -i 1 -k j j j -k -1 i -j k 1 -i k k j -i -1 -k -j i 1 -1 -1 -i -j -k 1 i j k -i -i 1 -k j i -1 k -j -j -j k 1 -i j -k -1 i -k -k -j i 1 k j -i -1
Red (i) and Blue (j) generators of  Q8

The real line combined with an  oriented  3-dimensional Euclidean space of orthonormal basis  (i,j,k)  forms the  quaternions,  a  4-dimensional  normed division algebra  similar to  2-dimensional  complex numbers, except multiplication is  not  commutative:

(a,A) + (b,B)	=	( a+b , A+B )
(a,A)  (b,B)	=	( ab - A.B  ,  aB + bA + A´B )

This is how the 3-dimensional "dot product" and "cross product" were  invented, well before the generalized idea of a vector became commonplace.

The above quaternionic units can be used to build a  Dirac operator  D  (yielding the opposite of the  Laplacian  D  when applied twice):

D    =    i ¶ / ¶x  +  j ¶ / ¶y  +  k ¶ / ¶z

The Laplacian remains the same in two systems of coordinates (a.ka. reference frames) obtained from each other by rigid rotation.

History of Quaternions (41:08)  by  Kathy Joseph  (Kathy Loves Physics & History, 2023-01-30).
 
Fantastic Quaternions (12:24)  by  James Grime  (Numberphile, 2016-01-18).
 
Visualizing quaternions, with stereographic projection (31:50)  Grant Sanderson  (3Blue1Brown, 2018-09-06).

(2023-03-10)   Gamma group  (order 32)  and spin group  (order 16).
The law for multiplication in the algebra generated by the  Dirac matrices.

The multiplicative group generated by the four gamma matrices a,b,c,d is of order 32.  It consists of 16 disjoint pairs of elements which are (additive) opposites of each other.  With one element of each such pairs, we form a basis for the algebra of dimension 16 generated by the 4 gamma matrices.

The group clearly contains the identity matrix of dimension four (I) and the product  e = abcd.  Introducing  e  allows every element of the group to be uniquely represented by a sign (+ 1 or -1) along with a signed product of at most two factors among the five elementary elements a,b,c,d,e.  With zero such factors, we have 2 elements (+I and -I),  with one factor we have 10 elements  (a,b,c,d,e and their opposites) and with 2 distinct factors, we obtain  20 = 2×C (5,2)  elements.  The grand total is indeed 32.

Among the 32 elements of the gamma group, we find:

• 1 element of order 1, namely I.
• 15 elements of order 2:  -I, a, -a, ab, -ab, ac, -ac, ad, -ad, ae, -ae.
• 20 elements of order 4, whose square is -I.

This algebra of dimension 16 is known as the  spacetime algebra  Cl (1,3)  which is just the  Clifford algebra  of dimension 4 with Minkowski metric.

Pseudoscalars (Grade 4) commute only with scalars or bivectors (Grade 2).  The bivectors and the scalars form the  centralizer  of the pseudoscalars.

The above table doesn't depend on Dirac's representation of  a,b,c,d  in terms of  4×4  matrices.  It can be entirely constructed from the pairwise anticommutativity of a,b,c,d and the following relations.  Therefore, an isomorphic group is entirely specified by a choice of a basis of four mutually anticommutative elements verifying these:

a² = I ,       b² = c² = d² = -I       and the definition   e = abcd

The last relation implies that  e² = -I   and also that  e  anticommutes with the other four.  Thus,  a  is special  (it's the only single-letter element which squares to unity)  but  b,c,d,e  are placed on an equal footing.

Here's one remarkable identity:

det ( ta + xb + yc + zd + ue)   =   ( t ² - x ² - y ² - z ² - u ²) ²

Enumeratimg the automorphisms of Dirac's gamma group :

Let  f  be an automorphism.  f (a)  must be of order 2 and is therefore,  up to a change of sign,  an element of  {a,ab,ac,ad,ae}.  For the three distinct images of b,c,d to anticommute with  f (a)  and with each other,  they must belong to  {b,c,d,e}  up to sign.  Conversely, if those conditions are met,  the images of a,b,c,d generate the whole group,  as the above table can be constructed using only the rules for combining single letters.  Thus,  we have  5  choices for  f (a)  and  4×3×2  choices for the other three letters,  knowing that we may then pick any choice of four signs among  16  possibilities.  Therefore:

Dirac's Gamma group has  5! 2⁴   =   120 × 16   =   1920   automorphisms.

(2014-12-17)   D₄ :  The fourth dihedral group  (8 elements)
The  octic group  is represented by the eight symmetries of a square.

This is a  centerless  group  G  isomorphic to  Aut(G)  but  not to  Inn(G).  A nice example of an  incomplete  group isomorphic to its automorphisms.

The dihedral group  D₄  can be represented as the group of the 8 symmetries of a square,  with vertices numbered clockwise  1,2,3,4.  It's generated by :

• r   =   (2341)   =   Quarter-turn clockwise  (order 4).
• s   =     (42)     =   Flip about a diagonal  (order 2).

Dihedral Group  D4    A   B   C   D   E   F   G   H   A  A B C D E F G H  B  B C D A H E F G  C  C D A B G H E F  D  D A B C F G H E  E  E F G H A B C D  F  F G H E D A B C  G  G H E F C D A B  H  H E F G B C D A		s r s   =   r -1   =   (1432)   A   =   e B   =   r C   =   r2 D   =   r3 E   =   s F   =   s r     =   r3 s G   =   s r2   =   r2 s H   =   s r3   =   r s
A  is the identity. C  is the half-turn. B and D  are quarter-turns. E and G  are diagonal flips. F and H  are side flips. Swapping an  even  number of the above pairs yields one of the  inner automorphisms  tabulated at right.		The 4 inner automorphisms  are all  even  permutations :    A   B   C   D   E   F   G   H   fA = fC  A B C D E F G H fB = fD A B C D G H E F fE = fG A D C B E H G F fF = fH A D C B G F E H

If there was an automorphism swapping an odd number of the three pairs (B,D), (E,G) and (F,H) then we could combine it with one of the four inner automorphisms to obtain some automorphism  f  leaving  (A,B,C,D)  invariant and swapping either (E,G) or (F,H).  Neither is possible, since:

• If  f  only swaps E and G, then   f (B)  f (F)  =  B F  =  E   ¹   f (B F)  =  G
• If  f  only swaps F and H, then   f (B)  f (E)  =  B E  =  H   ¹   f (B E)  =  F

Therefore, any other automorphism must involve sending at least one element of the three aforementioned pairs to an element of  another.

Any automorphism must leave invariant  A  (the identity)  and  C  (the only other element with a square root).  Likewise, the order-4 elements, B and D, must be invariant or transform into each other. 

Aut ( D₄ ) ,  the group of automorphisms of  D₄ ,  is isomorphic to  D₄.

One of the  8  isomorphisms between  D₄  and  Aut ( D₄ )

	D4			Aut ( D4 )			Inn ( D4 )
	A	1234	e		ABCDEFGH	a	fA = fC
	B	2341	r		ABCDFGHE	b
	C	3412	r2		ABCDGHEF	c	fB = fD
	D	4123	r3		ABCDHEFG	d
1	E	1432	s		ADCBEHGF	e	fE = fG
2	F	2143	s r  = r3s		ADCBHGFE	f
3	G	3214	s r2 = r2s		ADCBGFEH	g	fF = fH
4	H	4321	s r3 = r s		ADCBFEHG	h
Column 1 gives the correspondence between the square's vertices and the order-2 automorphisms,  which directly sends column 3 to column 5  (adjusting for even parity by swapping B and D if needed).

The  9  proper subgroups of  D₄  are abelian.  Seven of them are  cyclic  (one of order 1, five of order 2, one of order 4)  and two are Klein groups.

The  octic group  may also be  represented  as a group of  2 by 2  matrices:

A	B	C	D	E	F	G	H
1  0  0  1	0  1  -1 0	-1  0   0 -1	0 -1  1  0	0  1  1  0	-1  0  0  1	0 -1  -1  0	1  0  0 -1
e	r	r2	r3	s	s r	s r2	s r3

Automorphisms of the Dihedral Groups (1942)  by  George Abram Miller  (1863-1951).

(2006-05-09)   Enumeration of Groups of Small Order
The number g(n) of different groups of order n  (up to isomorphism).

If the integer  n  is coprime with its Euler totient  f(n), then there's only one group of order  n  (the cyclic group).  This applies to the following values of  n:  1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51... (A003277).  This result is attributed to William Burnside (1852-1927) and those numbers are known as  cyclic numbers.

A divisor of a  Carmichael number  is necessarily odd and cyclic.  Around 1980, I made the conjecture that the converse is true; any odd cyclic number seems to divide at least one Carmichael number  (if the conjecture is true, it divides infinitely many of them, since a cyclic number has infinitely many cyclic multiples).  In 2007, Joe Crump and myself proved this to hold for cyclic numbers  below 10000.  We're not attempting to gather more numerical evidence at this time...

For  noncyclic orders  (A060679)  here's the number of distinct groups:

Number of groups of order  n   (A000001)

n	g(n)	n	g(n)	n	g(n)	n	g(n)	n	g(n)	n	g(n)
4 6 8 9 10 12 14 16 18 20 21 22 24 25	2 2 5 2 2 5 2 14 5 5 2 2 15 2	26 27 28 30 32 34 36 38 39 40 42 44 45 46	2 5 4 4 51 2 14 2 2 14 6 4 2 2	48 49 50 52 54 55 56 57 58 60 62 63 64 66	52 2 5 5 15 2 13 2 2 13 2 4 267 4	68 70 72 74 75 76 78 80 81 82 84 86 88 90	5 4 50 2 3 4 6 52 15 2 15 2 12 10	92 93 94 96 98 99 100 102 104 105 106 108 110 111	4 2 2 231 5 2 16 4 14 2 2 45 6 2	112 114 116 117 118 120 121 122 124 125 126 128 129 130	43 6 5 4 2 47 2 2 4 5 16 2328 2 4

g(n) = 2   if  n  is  either  the square of a prime  or  a squarefree number with  only one  of its prime factors congruent to  1  modulo another  (A054395).  The following table gives, for each  m,  the numbers  n  for which  g(n) = m.

Smallest n for which there are exactly m groups of order n  (A046057):
1, 4, 75, 28, 8, 42, 375, 510, 308, 90, 140, 88, 56, 16, 24, 100, 675...

The convention would be to have a zero term to indicate that there's no such n but it's  conjectured  that this never happens.

Groups of order  2ⁿ  (A000679)   |   The Small Groups Library

(2006-03-05)   Classification of Finite Simple Groups   (1982)
The final result of the work of many group theorists over many years...

The finite simple  abelian groups  are just the cyclic groups of prime order.

The classification of  noncommutative  finite simple groups is  much  tougher...  Arguably, the final classification effort started with the 1963 publication of a 255-page proof of the  Odd Order Theorem  (or  Feit-Thompson theorem)  which implies that all noncommutative simple finite groups are of even order:

Solvability of Groups of Odd Order
by  John G. Thompson (1932-)  and   Walter Feit (1930-2004).
Pacific Journal of Mathematics  13  (1963)   775-1029.

The classification was declared complete in 1982,  despite pending gaps...  This was the result of a tremendous collective effort, spanning decades.  A key figure in this accomplishment was Daniel Gorenstein (1923-1992).

The Classification Theorem :

Unless it's one of the  27  sporadic groups  presented below  (including the  Tits Group,  often dubiously tallied with twisted Chevalley groups)  a  finite simple group  must belong to one of the following 18 countable families:

• The cyclic groups  C_p  of prime order  p.
• The alternating groups  A_n  of degree  n > 4   ( A₅  is of order  60 ).
• 16  types of Chevalley groups, listed below, each uniformly described in terms of a  finite field  of order  q  (q  being the power of a prime).  For example, the first such type consists of the  projective  group of square matrices of dimension  n+1  with coefficients in  F_q :

A_n(q)   =   PSL (n+1, F_q )

Simple Chevalley Groups   ( u Ù v   denotes the GCD of  u  and  v)

Symbol	Order
An(q) ;   n > 0 (q>3 if n=1)	qn(n+1)/2 (n+1) Ù (q-1) n (qi+1-1)    Õ    i =1
Bn(q) ;   n > 1 Except  B2(2)	qn2 2 Ù (q-1) n (q2i -1)    Õ    i =1
Cn(q) ;   n > 2
Dn(q) ;   n > 3	qn(n-1)  (qn-1) 4 Ù (qn-1) n-1 (q2i -1)    Õ    i =1
E6(q)	q36 (q12-1) (q9-1) (q8-1) (q6-1) (q5-1) (q2-1) / 3 Ù (q-1)
E7(q)	q63 (q18-1) (q14-1) (q12-1) (q10-1) (q8-1) (q6-1) (q2-1) / 2 Ù (q-1)
E8(q)	q120 (q30-1) (q24-1) (q20-1) (q18-1) (q14-1) (q12-1) (q8-1) (q2-1)
F4(q)	q24 (q12-1) (q8-1) (q6-1) (q2-1)
G2(q) Except  G2(2)	q6 (q6-1) (q2-1)
2An(q) ;   n > 1	qn(n+1)/2 (n+1) Ù (q+1) n (qi+1 - (-1)i+1 )    Õ    i =1
2B2(q) q = 2 2m+1 > 2	q2 (q2+1) (q-1)
2Dn(q) ;   n > 3	qn(n-1)  (qn+1) 4 Ù (qn+1) n-1 (q2i -1)    Õ    i =1
3D4(q)	q12 (q8+q4+1) (q6-1) (q2-1)
2E6(q)	q36 (q12-1) (q9+1) (q8-1) (q6-1) (q5+1) (q2-1) / 3 Ù (q+1)
2F4(q) q = 2 2m+1 > 2	q12 (q6+1) (q4-1) (q3+1) (q-1)
2G2(q) q = 3 2m+1 > 3	q2 (q2+1) (q-1)

Chevalley groups  are named after  Claude Chevalley (1909-1984)  who was the youngest founder of the Bourbaki group in 1935.

In 1955,  Chevalley  found a uniform way to describe  Lie  groups over arbitrary fields.  With  finite fields,  this led to what  J.H. Conway (1937-2020)  and others have called  untwisted  Chevalley groups  (they're listed first in the above table,  with unsuperscripted symbols).

The  twisted  Chevalley groups  (denoted by superscripted symbols)  result from two modifications of Chevalley's approach.  One was proposed in 1959 by  Robert Steinberg (1922-2014).  The other (1960-1961) is due to  Michio Suzuki (1926-1998)  and  Rimhak Ree (1922-2005).

The above  highlighted entry  ²F₄(2 ^2m+1 )  is  simple  only for positive values of m.  For m=0,  this group is not simple but it has a simple normal subgroup of index 2  and order  17971200  (its derived subgroup)  which is known as the  Tits Group,  and is best classified among  sporadic  groups.

Classification of Finite Simple Groups   |   List of Finite Simple Groups
 
Jordan-Hölder theorem   |   Camille Jordan (1838-1922; X1855)   |   Otto Hölder (1859-1937)
 
Simple Groups (8:52)  by  Liliana De Castro  (Socratica, 2017-01-10).

(2006-03-06)   The 26 or 27 Sporadic Groups
Noncommutative non-alternating  finite simple groups  not of Lie type.

20 of these are related to the largest and most famous of them all,  the  Fischer-Griess Monster.  Six other sporadic groups ( highlighted ) unrelated to the  Monster  are known as  oddments  or  pariahs.

The  27^th sporadic group is, arguably, the  aforementioned  Tits Group.

Sporadic Notes :

The Mathieu group  M₂₁  doesn't belong to the above list.  It's  simple  but can't be considered  sporadic  because it's isomorphic to PSL(3,4):

M₂₁   =   PSL(3,4)   =   PSL(3,F₄ )   =   A₂ (4)

The  Fischer-Griess Monster Group  is also known as  Fischer's Monster  or the  Monster Group.  It was predicted independently by Bernd Fischer and Robert L. Griess in 1973.  At first, Griess dubbed it the  Friendly Giant  and constructed it explicitely in 1981, as the automorphism group of a 196883-dimensional commutative nonassociative algebra over the rational numbers.

The  Leech Lattice  is the densest packing of 24-dimensional hyperspheres  (each touches 196560 others).  Its automorphisms feature a center of order two.  Modulo that center, they form the  Conway Group  (Co₁).

Simon P. Norton gave a construction of the group proposed by Koichiro Harada  (now called the Harada-Norton group).  Norton also proposed the  monstruous moonshine conjecture  with his advisor, John H. Conway.

The  Higman-Sims Group  (HS)  is named after Donald G. Higman and Charles C. Sims, who described it jointly in 1968.  It's a subgroup of index 2 in the group of automorphisms of the  Higman-Sims graph  (the strongly-regular graph with 100 nodes of degree 22, where adjacent nodes have no common neighbors and nonadjacent nodes have 6 common neighbors).

The  Hall-Janko Group  (HJ)  is named after  Marshall Hall, Jr. (1910-1990)  and  Zvonimir Janko (1932-).  It's a subgroup of index 2 in the automorphisms of the  Hall-Wales graph  constructed in 1968 by  David Wales,  as the strongly-regular graph with 100 nodes of degree 36, where adjacent nodes have 14 common neighbors and nonadjacent nodes have 12  (also called Hall-Janko graph).

The modern quest for a complete list of sporadic groups was launched by the discovery of the first of the Janko Groups  (J₁) by  Zvonimir Janko,  in 1965.

The first sporadic groups  (M₁₁ , M₁₂ , M₂₂ , M₂₃ , M₂₄ )  are subgroups of  M₂₄   discovered between 1860 and 1873 by Emile Mathieu (1835-1890; X1854).  Georg Frobenius (1849-1917)  proved  M₁₂  to be simple in 1904.

The Mathematicians Involved   |   Monstrous Moonshine Theory  (Wikipedia)
 
Video :   Finite groups, Yesterday and Today  by  Jean-Pierre Serre  (Harvard, 2015-04-24).

(2017-08-02)   Torsion  T = Tor(G)  of an infinite group  G
T  is the set of all elements of  G  which have a finite order.

An element of finite order is called a  torsion element.  If the identity is the only such element,  the group  G  is said to be  torsion-free.

A torsion element whose order divides  k  is called a  k-torsion.

On the other hand,  a  torsion group  (also called a  periodic group)  is a group consisting only of torsion elements  (which is to say that all elements have finite orders).  All finite groups are periodic  (i.e.,  Tor(G) = G).  If the orders of the elements in a periodic group are bounded,  then they have a  least common multiple  n  and the group is said to be  of exponent  n.

One example of an  infinite  finitely-generated torsion group was given in 1964,  by  Evgeny Golod (1935-2018)  and  Igor Shafarevich (1923-2017).

Burnside problem (1902)   |   William Burnside (1852-1927)

(2015-05-03)   Linear representations of a group  G :
Homomorphisms from G into a group of matrices.

GL(n,K)  is the group of invertible n by n matrices with entries in a field K.

All finite groups are linear. 
Compact groups...
Lie groups...
Faithful representations (isomorphisms).
Irreducible representations do not allow any nontrivial proper invariant subspace.

Shur's lemma   |   Issai Schur (1875-1941)
 
Representation Theory Basics  by  Robert Donley   (2011-03-08).

(2023-04-04)   Lie Groups.
Groups which are also smooth manifolds  (locally Euclidean).

The tangent space to a Lie group is a  Lie algebra.  The converse is true in finitely many dimensions but there are Lie algebras with infinitely many dimensions which cannot be realized as the tangent space to a Lie group.  The  earliest counterexample  is due to the  bourbakist  Adrien Douady (1935-2006).

Lie group

(2006-03-01)   Classical Groups   (multiplicative subgroups of matrices)
Groups of transformations depending on parameters in a  field.

The classical groups tabulated below are subgroups of the group  GL(n,K)  of invertible n by n matrices with entries in the field K.

When  K  isn't specifed, the field of real numbers  (R)  is understood, except that the field of complex numbers  (C)  underlies the groups denoted  U(n)  and  SU(n)  (note, however, that the "dimension" listed is always the real dimension, which is twice the complex dimension whenever applicable).

A subgroup of  GL(n,K)  is called a  linear representation  (or simply a  representation)  of any group it happens to be isomorphic to.

A*  denotes the  adjoint  of the square matrix  A  (namely, the "conjugate transpose" of a complex matrix, or simply the transpose of a real matrix).

A matrix is said to be  unimodular  if its  determinant  is 1.  In the symbol of a group, the letter "S" (for  special) says that its elements are unimodular.

Alternate Notations :

A notation like  GL(Kⁿ)  may also be used instead of  GL(n,K).  This has the great advantage of being consistent with more general symbols like  GL(V)  which apply to a vector space  V  whose dimension  may  be infinite.

On the other hand, when a finite field  is used, GL(n,GF(q))  may be denoted  GL(n,q).  A similar convention holds for all the symbols tabulated above.  For example, the first type of Chevalley groups is  PSL(n,q) = A_n(q).

There's no risk of confusion with notations like  O(3,1)  as used below, which refer to a real vector space metrically endowed with 3 spacelike dimensions and 1 timelike dimension,  since we've yet to conceive several dimensions of time  and rarely consider a field of one element.

Some Special Cases :

• The simplest unitary group is the "unit circle" or  circle group  (denoted T)  which is isomorphic to  U(1),  SO(2)  and    / .
• SZ(n,C)  is the cyclic group of order n  (it does "look" cyclic).
• The  Möbius Group  is isomorphic to  PGL(2,C)  and/or  PSL(2,C).

Classical groups   |   "The Classical groups" (1939)  by  Hermann Weyl (1885-1955)

(2020-09-22)   Lie Groups
Groups of  diffeomorphisms  over a  smooth manifold.

With  K = (or )  the above classical groups  are examples of  Lie groups.

Lie Algebra of a Lie group
 
Lie groups and their Lie algebras (1:43:11)  by  Frederic P. Schuller  (IAS, 2015-09-21).

(2016-05-21)   Projective Group
Linear group modulo the scalar group or any group modulo its center.

Traditionally, the  projective group  is the quotient of the  general linear group  (i.e., the group of all square matrices of a given dimension over a given field)  modulo the  scalar group  (i.e., the diagonal matrices).

The term is also used as a qualifier to denote the quotients nodulo the scalar group of some subgroups of the general linear group.

By extension, the qualifier  projective  can even be used to denote the quotient of any group modulo its own center.  (See  modular group.)

The qualifier "projective" is inherited from the name given to the rules of geometrical perspective, first devised by Renaissance artists.  In their drawings, they mapped every point (P) of three-dimensional Euclidean space to the unique point (M) of a planar canvas intersecting the straight line (OP) drawn from that point to the eye of the observer (O).  In such a mapping, a horizontal plane is mapped onto a half-plane of the canvas which ends on a straight line representing the horizon  (supposedly "at infinity").
 
In one of the greatest leaps of imagination ever made by the human mind, the geometers of the nineteen century realized that this artistic rendering was just a special case of the above and they would eventually turn projective geometry into a very fruitful independent field of study.  Once called  higher geometry,  that became a revered part or higher learning before being all but forgotten...

Projective space

(2006-04-12)  The Möbius Group  (homographic transformations)
The automorphisms of the  Riemann Sphere  (the  projective line).

An  homographic transformation  f  (also called a  Möbius transformation  or a  fractional linear transformation)  sends a complex number  z  to:

f (z)   =	a z  +  b
	c z  +  d

It's a  [bijective]  transformation  of the  projective line  (the complex plane plus a single "infinity" point  ¥  beyond its horizon, so to speak).  The image of  ¥  is  a/c  (or  ¥  if  c = 0 ).  The image of -d/c  (or  ¥  if  c = 0 )  is  ¥.

The Stereographic Projection

Projective Line	Riemann Sphere
È {¥}	(a,b,c)  Π 3   |   a 2 + b 2 + c 2  =  1
¥	(0,0,1)
z   =     a  +  i b    1 - c	(a,b,c)     c ¹ 1
z  =  u + iv	(   2 u   ,   2 v   ,   | z | 2 - 1   )   | z | 2 + 1 | z | 2 + 1 | z | 2 + 1

Automorphic functions  (originally dubbed "Fuchsian functions" by Poincaré, around 1884)  are  meromorphic functions  (i.e., ratios of two  holomorphic functions;  analytic functions of a complex variable)  which are invariant under a countable infinity of Möbius transformations).

Moebius Transformations Revealed  (a great video)   by  Douglas N. Arnold  &  Jonathan Rogness

(2016-05-22)  The Modular Group  G
The common name of the  projective special linear group   PSL(2,).

The locution  gamma group  is best reserved for something else.

The  modular group  consists of all  2 by 2  square matrices with  integer  elements  (in )  and  unit  determinant  (that's what  special  means)  when considered  modulo  the  center  {I,-I}  (that's what  projective  means).

That last specification merely states that a matrix and its opposite are equivalent representations of the same element of the  modular group.

The  modular group  G  has the following presentation:

G   =   < S, T  |  S² , (ST)³ >

G  is a discrete subgroup of the  Möbius group,  represented as follows:

Name  f	S	T	Tn	STn	Tn S
± matrix	0 -1   1  0	1  1   0  1	1  n   0  1	0 -1   1  n	n -1   1  0
f (z)	-1/z	z+1	z+n	-1 z+n	n-1/z

The  modular group  was first studied in detail, for its own sake, by  Richard Dedekind  and  Felix Klein  as part of the  Erlangen program  (1872).  The closely related  elliptic functions  (introduced by  Lagrange  in 1785)  had already been studied quite extensively by  Abel (1827-1828)  and  Jacobi (1829)  who shared  the  grand prix  of the French  Academy of Sciences  for that work,  in 1830  (after Abel's death).

An interesting source of examples in the modular group is provided by the successive convergents obtained by truncating the  continued fraction expansion of a number, because the following relation is naturally satisfied:

P_n+1 Q_n  -  Q_n+1 P_n   =   (-1)ⁿ

Elliptic functions, modular forms, Hecke theory, etc. & theta functions   by  Ben Brubaker  (MIT, 2008).
Generating the modular group  (Malik Younsi, 2010)   &   Euclidean algorithm  (Qiaochu Yuan, 2008).
The modular group and words in its two generators  by  Giedrius Alkauskas  (2016-04-10)   A265434
The Modular Group ... subgroups  McCreary, Murphy, Carter.  The Mathematica Journal, 9, 3  (2005).
Geometry and Groups  by  T. Keith Carne, University of Cambridge  (2012).
The Modular Group and its Actions  by  A. Muhammed Uludag,  with Appendix by Hakan Ayral  (2013).
The Minkowski Question Mark and the Modular Group  by  Linas Vepstas  (2014).
 
Wikipedia :   Modular group   |   Moduli spaces

(2017-07-29)   Group Structure of an  Elliptic Curve
Group operator defined on a cubic planar curve without singular points.

In the Euclidean plane,  a cubic curve without singular points is called an  elliptic curve.  That same term is also commonly used to denote the cartesian equation of such a curve or the wonderful group structure its points can be endowed with,  as described below.  Elliptic curves can be considered over various  fields  (complex numbers, rationals, p-adic numbers, finite fields).

Elliptic curves over  finite fields  allow elliptic-curve cryptography (ECC)  which was invented in 1985 and has been widely used since 2004.

Mordell's Theorem  (1922) :

In 1901,  Poincaré  had asked whether the rational points of a curve of genus 1 are finitely generated.  21 years later, Mordell  settled that for elliptic curves:

An elliptic-curve's rational points form a finitely-generated abelian group.

For an elliptic curve E,  this is denoted  E() 

Elliptic curves   |   Group law of an elliptic curve   |   Addition theorems   |   Finitely generated abelian group
EEC:  Elliptic-curve cryptography (1985)   |   Neal Koblitz (1948-)   |   Victor S. Miller (1947-)
Birch and Swinnerton-Dyer conjecture  about the rank of an elliptic curve.

(2017-08-03)   Group Law on a Degenerate Cubic Curve
Combining a circle and a straight line so the latter is a subgroup.

In the Euclidean plane,  let's apply the geometric definition of  sums on an elliptic curve  to the degenerate cubic consisting of a circle of unit diameter and a straight line at a distance  d  from its center.

When at least one point is on the circle,  the geometric construction of the sum of two points presents no difficulty.  On the other hand,  if both of the points A and B are on the line,  their sum  C = A+B  is not immediately clear.  To construct it,  we may consider any auxiliary point V on the circle and use the following identity, involving three sums of the previous kind:

A + B   =   ( (A+V) + B) - V

For convenience,  we choose V on the axis of symmetry of the figure,  so that  V = -V,  in which case we have a symmetrical defining relation:

A + B   =   (A+V) + (B+V)

If  A'  is the mirror-image of  A+V  (with respect to the horizontal axis of symmetry)  then the law introduced in  the non-degenerate case  says that  A'  is at the intersection of the circle and the AV line.  Likewise,  the image  B'  of  B+V  is the intersection of  BV  with the circle.  A+B  is on the mirror-image of the line joining  A+V  and  B+V,  which is the line A'B'.  So,  A+B  is at the intersection of  A'B'  with our basic vertical line,  as shown in the figure at left.

w   =        u + v  1  +  k uv

If we're concerned with number theory,  we choose any rational value for  k.  Otherwise,  we remark that the above equation encodes a group structure on the real line in one of three different ways,  modulo some rescaling:

• k  =  0.  Ordinary addition.
• k  =  -1.  Addition of trigonometric tangents.
• k  =  1 / c². Addition of hyperbolic tangents  (relativistic  rapidities).

Moreover,  the limiting case when  k  tends to infinity can be construed as ordinary multiplication of the reciprocals of nonzero numbers.  Of course, the (nonzero) rational numbers are not finitely generated under this law,  because there are  infinitely many prime numbers.

More generally,  we may consider any continuous monotonous function  f  from negative infinity to positive infinity and define an abelian group law over the real numbers by:

x o y   =   f (  f ^-1 (x)  +  f ^-1 (y) )

Our previous discussion is a special case of that if we choose  f  to be either the trigonometric tangent or the hyperbolic tangent.  The former for a line which doesn't intersect the basic circle, the latter for a line which does.

Relativistic addition of parallel velocities   |   Broken-calculator puzzle

(2017-07-30)   Amenable Groups   (French:  groupes moyennables)
Introduced by  Von Neumann  to discuss the Banach-Tarski paradox.

An  amenable group  is a  locally compact  topological group whose elements leave invariant some kind of averaging on bounded functions.

The English word was coined in 1949 by  Mahlon M. Day  as a pun  ("a-mean-able")  to translate the German term originally used by Von Neumann in 1929  (messbar = measurable).  The French use either the English term or the (better) word  moyennable.

Amenable group

(2017-07-28)   Richard J. Thompson's Groups   (1965)
F is the smallest of the three nested groups  F,  T  and  V.

The three Thompson groups  F,  T  and  V  are also called  vagabond groups,  chameleon groups  or just  chameleons  (the latter term was coined by  Matt Brin  in 1994).  They have unusual properties which have made them counterexamples to several conjectures in group theory.

F is not  simple  but its derived group is.  T and V are simple.  T was the first known example of a   finitely-presented infinite simple group.

Group F :

F can be defined as the subgroup of the piecewise-linear automorphisms of the interval  [0,1]  consisting of all functions  f  such that:

• f  is differentiable outside of a finite set of  dyadic  rational numbers  (i.e.,  values of fractions whose denominators are powers of 2).
• When it exists,  the derivative of  f  is a power of 2.

Group T :

Group V :

Thompson groups  (F, T and V)   |   Richard J. Thompson (1933-2015?) Ph.D. 1979)   |   Matthew G. Brin
The Higman-Thompson groups  by  Peter Cameron  (2016-06-05)   |   Graham Higman (1917-2008)
The many faces of Thompson's Group F  by  John Meier  (2004-09)
What is Thompson's Group?  by  J.W. Cannon & W.J. Floyd  (2011-08)
Presentations of Thompson's group V by permutations  by  Collin Bleak & Martyn Quick  (2015-11-06).
 
An introduction to Thompson's group F (and T) for physicists (33:32)  by  Tobias Osborne  (2016-01-27).
The chameleon groups of Richards J. Thompson  by  Matthew G. Brin  (IHES, 84, pp. 5-33, 1996).

(2020-01-27)  Abelian sandpiles and sandpile groups

Let M be a sandpile for which there's a sandpile Z such that  M+Z = M.  Then, Z is a zero  (i.e. it's a neutral element for addition)  over the set of all sandpiles of the form X+M, since:

(X+M) + Z   =   X + (M+Z)   =   X+M

Diffusion limited aggregation (DLA, 1981)   |   Thomas A. Witten, Jr. (1944-)   |   Leonard M. Sander (1941-).
Abelian sandpiles (1987)   |   Per Bak (1948-2002)   |   Tang Chao (1958-)   |   Kurt Wiesenfeld (1958-)
 
The Abelian Sandpile and Related Models  by  Deepak Dhar  (1998-10-22).
The Abelian sandpile; a mathematical introduction  by  R. Meester, F. Redig, D. Znamenski  (2001-06-12).
What is a Sandpile?  by  Lionel Levine & James Propp  (Notices of the AMS, 57, 8, pp. 976-979, Sept. 2010).
Primer for the Algebraic Geometry of Sandpiles  by  D. Perkinson, J. Perlman, & J. Wilmes  (2012-01-04).
 
Sandpile Fractal Generation (1:16:29)  by  Harrison Lambeth  (2017-02-01).
Convergence of the Abelian sandpile (57:48)  by  Charles Smart  (Microsoft Research, 2016-04-16).
Self-Organized Criticality (10:01)  by  Andrew Hoffman  (2015-05-12).
Sandpiles (24:09)  by  Luis David Garcia-Puente  (Numberphile, 2017-01-13).

(2006-03-01)  The Lorentz Group  O(3,1)  has 4 connected components.
Each is isomorphic to the  Restricted Lorentz Group  SO⁺(3,1).

h   =

é ê ê ë

-1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

ù ú ú û

The  Lorentz Group  O(3,1)  is isomorphic to  SL(2,C)  and consists of all  4 by 4  real matrices  A  such that A* h A = 1, where h is the metric matrix for three dimensions of space and one dimension of time.

The  O(3,1)  group has  4  connected components.  Those components are pairwise homeomorphic and are  not  simply connected :

SO⁺(3,1)       T[ SO⁺(3,1) ]       P[ SO⁺(3,1) ]       PT[ SO⁺(3,1) ]

SO⁺(3,1)  is the  (6-dimensional)  Restricted Lorentz Group  consisting of the elements of the  Lorentz Group  O(3,1)  which preserve the direction of time and the orientation of space  (boosts and 3D rotations).  In the above, T and P denote a reversal of time and an inversion of space  (the latter could be either a mirror symmetry about a plane or a symmetry about a point).

The symbol  SO(3,1)  would denote the "Special Lorentz Group", the subgroup of the matrices of  O(3,1)  with determinant one  (which is a disconnected "half" of  O(3,1),  not a connected "quarter" of it).

Poincaré Group :

The  Poincaré Group  ISO⁺(3,1)  is the  10-dimensional  inhomogeneous  group of noninverting isometries for 3 dimensions of space and one dimension of time.  It consists of transformations mapping  x  to  Lx+a , where  L  belongs to the above  Restricted Lorentz Group  SO⁺(3,1)  and  a  is some  4-vector.

Wigner's Classification :

Wikipedia :   Wigner's classification   |   Wigner's theorem (1931)   |   Eugene Wigner (1902-1995; Nobel 1963)
Baker-Campbell-Hausdorff formula

(2006-03-21)   Local Symmetries of the Physical Universe :  A Primer
The laws of nature are invariant under a certain group of transformations.

God does arithmetic.
Carl Friedrich Gauss  (1777-1855) 

In spite of their respective successes, General Relativity and the Standard Model are known to be imperfect theories, incompatible with each other.  The ultimate laws of physics  (if they exist)  could only incorporate those two as approximations applicable to specific experimental domains  (like  Newtonian mechanics  approximates  Special Relativity  for low speeds).

Nobody knows (yet) exactly what symmetries the  ultimate laws  of nature should have,  but we may ponder the groups of local symmetries underlying modern mathematical theories of the  4  known physical interactions:

Force	Group	Dimension
Electromagnetism	U(1)	1
Weak interactions	SU(2)	3
Strong interactions	SU(3)	8
Gravity	ISO+(3,1)	10

Maxwell's unification of electricity and magnetism into electromagnetism has been ultimately  construed  as the discovery that electrodynamics is invariant under local  phase transformations,  with the simple structure of  U(1).  The classical quantity associated with that symmetry  (by  Noether's theorem )  is simply  electric charge.

Quantum electrodynamics (QED)  describes electromagnetism as a quantum field.  It became the basic paradigm for all subsequent quantum theories of fundamental physical interactions.  QED describes how  photons  "mediate" the force between  electrons  (or any other charged particles).

The  electroweak theory  is a satisfying unification of electromagnetism and weak interactions under the symmetries of the  direct product  SU(2)´U(1).  It was devised in 1967 by  Steven Weinberg  (1933-)  and  Abdus Salam  (1926-1996)  building on earlier work of  Sheldon Glashow  (1932-).  The three men shared the  1979 Nobel prize  for this.  The group  SU(2)  is isomorphic to 3-dimensional rotations.  The  broken  electroweak symmetry translates into  4  vector bosons:   g  (the photon)  Z⁰,  W⁺  and  W^-.

Broken:  In mathematical physics, a symmetry is said to be broken when symmetrical equations have an asymmetrical solution.

The theory of strong interactions is known as  quantum chromodynamics  (QCD).  It's based on an  unbroken  SU(3)  local symmetry, dubbed  color symmetry  because of a superficial similarity with the rules of color vision  (whereby 3 primary colors may combine to create  colorlessness).  QCD describes how  gluons  mediate the strong force between  quarks  (or anything else with  color charge, including gluons themselves).  There are 8 different types of gluons, corresponding to the 8 dimensions of SU(3).  In this context,  SU(3)  is often denoted  SU_c(3).  "C" stands for color.

As described by  Albert Einstein's  General Theory of Relativity,  gravity's local symmetry is that of the Poincaré group, which preserves spacetime intervals, as well as the direction of time and the orientation of space.  The  Poincaré group  is 10-dimensional.  However, a gauge field  (the  graviton)  is associated only with the 4 dimensions of spacetime translations.  Suspiciously, no such particle or field is associated with the 6 dimensions corresponding to Lorentz symmetries  (3 dimensions for spatial rotations and 3 dimensions for Lorentz boosts).

The so-called  Standard Model  of particle physicists describes both  strong  and  electroweak  interactions in a theoretical framework whose symmetries are those of the group  SU(2)´U(1)´SU_c(3), which has 12 dimensions.

The model depends on several parameters, adjusted to fit experimental data but otherwise unexplained.  Different local symmetries would impose different restrictions, for better or for worse.  One classical group possessing more dimensions of symmetry  (24)  than the  Standard Model  is  SU(5).

The correct local symmetry of  "strong-electroweak"  interactions would still not determine the masses of the vector bosons involved  (particles of spin 1)  unless more is known about the way such a symmetry is  broken.

A key aspect of particle physics which is based on a broken symmetry is the classification of elementary particles into three generations of  flavors.

A mind-boggling  supersymmetry  across different spins  (SUSY)  seems required of any quantum theory  designed  to include gravity in a fully unified quantum theory "of everything":  Supergravity, Superstrings, etc.

In 2010,  Sir Michael Atiyah (1929-2019)  remarked that the known physical symmetries occur naturally in the Tits-Freudenthal magic square pertaining to associative hypercomplex numbers obtained through the  Cayley-Dickson construct.  He speculated that the introduction of the fourth (nonassociative) hypercomplex division algebra  (the octonions)  is somehow related to gravity.  He admonished younger investigators to consider this possibility,  which would give a beautiful role to  all  exceptional Lie groups.

Yang-Mills theory (1954)   |   Weak interactions   |   Strong interactions   |   String theory
Pauli matrices  for SU(2)   |   Gell-Mann matrices  for SU(3)   |   Generalizations
Grand Unified Theories :  SU(5),  SO(10) ...
 
Yang-Mills Theories (3:21)  by  Murray Gell-Mann  (1998).
Mirror Symmetry & Geometric Langlands (1:11:21)  by  Ed Witten  (2012-10-18).
Les Symétries de l'univers (French, 15:34)  by  Alessandro Roussel  (ScienceClic, 2021-02-06).
 
Dirac's belt trick, Topology, and Spin ½ (59:42)  by  Noah Miller  (2021-08-23).

(2019-03-12)   Renormalization Group and Cosmic Galois Group
QFT's  renormalization group  is a subgroup of the  cosmic Galois group.

The name  cosmic Galois group  was coined by  Pierre Cartier  around 1998,  as he shared the optimism of Fields medalist  Maxim Kontsevich (1964-)  in the following words:

La parenté de plus en plus manifeste entre le groupe de Grothendieck-Teichmüller (GT) d'une part, et le groupe de renormalisation de la Théorie Quantique des Champs n'est sans doute que la première manifestation d'un groupe de symétrie des constantes fondamentales de la physique,  une espèce de  groupe de Galois cosmique !

The subject caught the attention of several other people connected with the  IHES.  One comprehensive introduction appears in the  work (2004)  of  Alain Connes (1947-)  and  Matilde Marcolli (1969-).

Grothendieck-Teichmüller group (Drinfeld, 1990)   |   Absolute Galois group (of the rationals)
Vladimir Drinfeld (1954-)   |   Alexander Grothendieck (1928-1914)   |   Oswald Teichmüller (1913-1943)
 
nLab
 
Cosmic Galois Group  by  Pierre Cartier  (Lectorium, Euler Institute, 2013-10-21).