Normal closure (group theory)

Formally, if ${\displaystyle G}$  is a group and ${\displaystyle S}$  is a subset of ${\displaystyle G,}$  the normal closure ${\displaystyle \operatorname {ncl} _{G}(S)}$  of ${\displaystyle S}$  is the intersection of all normal subgroups of ${\displaystyle G}$  containing ${\displaystyle S}$ :^[1] $${\displaystyle \operatorname {ncl} _{G}(S)=\bigcap _{S\subseteq N\triangleleft G}N.}$$ 

The normal closure ${\displaystyle \operatorname {ncl} _{G}(S)}$  is the smallest normal subgroup of ${\displaystyle G}$  containing ${\displaystyle S,}$ ^[1] in the sense that ${\displaystyle \operatorname {ncl} _{G}(S)}$  is a subset of every normal subgroup of ${\displaystyle G}$  that contains ${\displaystyle S.}$ 

The subgroup ${\displaystyle \operatorname {ncl} _{G}(S)}$  is generated by the set ${\displaystyle S^{G}=\{s^{g}:g\in G\}=\{g^{-1}sg:g\in G\}}$  of all conjugates of elements of ${\displaystyle S}$  in ${\displaystyle G.}$ 

Therefore one can also write $${\displaystyle \operatorname {ncl} _{G}(S)=\{g_{1}^{-1}s_{1}^{\epsilon _{1}}g_{1}\dots g_{n}^{-1}s_{n}^{\epsilon _{n}}g_{n}:n\geq 0,\epsilon _{i}=\pm 1,s_{i}\in S,g_{i}\in G\}.}$$ 

Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set ${\displaystyle \varnothing }$  is the trivial subgroup.^[2]

A variety of other notations are used for the normal closure in the literature, including ${\displaystyle \langle S^{G}\rangle ,}$  ${\displaystyle \langle S\rangle ^{G},}$  ${\displaystyle \langle \langle S\rangle \rangle _{G},}$  and ${\displaystyle \langle \langle S\rangle \rangle ^{G}.}$ 

Dual to the concept of normal closure is that of normal interior or normal core, defined as the join of all normal subgroups contained in ${\displaystyle S.}$ ^[3]

For a group ${\displaystyle G}$  given by a presentation ${\displaystyle G=\langle S\mid R\rangle }$  with generators ${\displaystyle S}$  and defining relators ${\displaystyle R,}$  the presentation notation means that ${\displaystyle G}$  is the quotient group ${\displaystyle G=F(S)/\operatorname {ncl} _{F(S)}(R),}$  where ${\displaystyle F(S)}$  is a free group on ${\displaystyle S.}$ ^[4]