Normal closure (group theory)

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In group theory, the normal closure of a subset ${\displaystyle S}$of a group ${\displaystyle G}$is the smallest normal subgroup of ${\displaystyle G}$containing ${\displaystyle S.}$

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 ${\displaystyle S}$: $${\displaystyle \operatorname {ncl} _{G}(S)=\bigcap _{S\subseteq N\triangleleft G}N.}$$

The normal closure ${\displaystyle \operatorname {ncl} _{G}(S)}$is the smallest normal subgroup of ${\displaystyle G}$containing ${\displaystyle S,}$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 the subgroup generated by the set ${\displaystyle S^{G}=\{s^{g}:s\in S,g\in G\}=\{g^{-1}sg:s\in S,g\in G\}}$of all conjugates of elements of ${\displaystyle S}$in ${\displaystyle G.}$Therefore one can also write the subgroup as the set of all products of conjugates of elements of ${\displaystyle S}$or their inverses: $${\displaystyle \operatorname {ncl} _{G}(S)=\{g_{1}^{-1}s_{1}^{\epsilon _{1}}g_{1}\cdots g_{n}^{-1}s_{n}^{\epsilon _{n}}g_{n}:n\geq 0,\epsilon _{i}=\pm 1,s_{i}\in S,g_{i}\in G\}.}$$

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

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 ${\displaystyle S.}$

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 ${\displaystyle S.}$