Normal closure (group theory)

This article is about the normal closure of a subset of a group. For the normal closure of a field extension, see Normal closure (field theory).

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 ${\displaystyle S.}$

Properties and description

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 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.^[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]

Group presentations

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]