Normal closure (group theory)

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

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

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

The subgroup ${\displaystyle \operatorname {ncl} _{G}(S)}${\displaystyle \operatorname {ncl} _{G}(S)} is the subgroup generated by the set ${\displaystyle S^{G}=\{s^{g}:s\in S,g\in G\}=\{g^{-1}sg:s\in S,g\in G\}}${\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 ${\displaystyle S}${\displaystyle S} in ${\displaystyle G.}${\displaystyle G.} Therefore one can also write the subgroup as the set of all products of conjugates of elements of ${\displaystyle S}${\displaystyle S} or their inverses: $${\displaystyle \operatorname {ncl} _{G}(S)=\{g_{1}^{-1}s_{1}^{\epsilon _{1}}g_{1}\cdots g_{n}^{-1}s_{n}^{\epsilon _{n}}g_{n}:n\geq 0,\epsilon _{i}=\pm 1,s_{i}\in S,g_{i}\in G\}.}$${\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 ${\displaystyle \varnothing }${\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^{G}\rangle ,} ${\displaystyle \langle S\rangle ^{G},}${\displaystyle \langle S\rangle ^{G},} ${\displaystyle \langle \langle S\rangle \rangle _{G},}${\displaystyle \langle \langle S\rangle \rangle _{G},} and ${\displaystyle \langle \langle S\rangle \rangle ^{G}.}${\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 ${\displaystyle S.}${\displaystyle S.}^[3]

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

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