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Normal closure (group theory)

(Redirected from Conjugate closure)

In group theory, the normal closure of a subset of a group is the smallest normal subgroup of containing

Properties and description

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Formally, if   is a group and   is a subset of   the normal closure   of   is the intersection of all normal subgroups of   containing  :[1]  

The normal closure   is the smallest normal subgroup of   containing  [1] in the sense that   is a subset of every normal subgroup of   that contains  

The subgroup   is generated by the set   of all conjugates of elements of   in  

Therefore one can also write  

Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set   is the trivial subgroup.[2]

A variety of other notations are used for the normal closure in the literature, including       and  

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  [3]

Group presentations

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For a group   given by a presentation   with generators   and defining relators   the presentation notation means that   is the quotient group   where   is a free group on  [4]

References

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  1. ^ a b Derek F. Holt; Bettina Eick; Eamonn A. O'Brien (2005). Handbook of Computational Group Theory. CRC Press. p. 14. ISBN 1-58488-372-3.
  2. ^ Rotman, Joseph J. (1995). An introduction to the theory of groups. Graduate Texts in Mathematics. Vol. 148 (Fourth ed.). New York: Springer-Verlag. p. 32. doi:10.1007/978-1-4612-4176-8. ISBN 0-387-94285-8. MR 1307623.
  3. ^ Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. p. 16. ISBN 0-387-94461-3. Zbl 0836.20001.
  4. ^ Lyndon, Roger C.; Schupp, Paul E. (2001). Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin. p. 87. ISBN 3-540-41158-5. MR 1812024.