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 $S$ of a group $G$ is the smallest normal subgroup of $G$ containing $S.$

Properties and description

Formally, if $G$ is a group and $S$ is a subset of $G,$ the normal closure $\operatorname {ncl} _{G}(S)$ of $S$ is the intersection of all normal subgroups of $G$ containing $S$:

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

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

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

Therefore one can also write

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

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

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

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 $S.$

Group presentations

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