Normal closure (group theory)

Template:Short description Template:About Template:Sidebar with collapsible lists 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 ${\mathrm{ncl}}_G(S)$ of $S$ is the intersection of all normal subgroups of $G$ containing $S$: $${\mathrm{ncl}}_G(S)=\bigcap _{S\subseteq N◃G}N.$$

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

The subgroup ${\mathrm{ncl}}_G(S)$ is the subgroup generated by the set $S^G=\{s^g:s\in S,g\in G\}=\{g^{-1}sg:s\in S,g\in G\}$ of all conjugates of elements of $S$ in $G.$ Therefore, one can also write the subgroup as the set of all products of conjugates of elements of $S$ or their inverses: $${\mathrm{ncl}}_G(S)=\{g_1^{-1}{s}_1^{{ϵ}_1}{g}_1\cdots g_n^{-1}{s}_n^{{ϵ}_n}{g}_n:n\ge 0,{ϵ}_i=\pm 1,s_i\in S,g_i\in G\}.$$

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

A variety of other notations are used for the normal closure in the literature, including $⟨S^G⟩,$ $⟨S{⟩}^G,$ $⟨⟨S⟩{⟩}_G,$ and $⟨⟨S⟩{⟩}^G.$

Dual to the concept of normal closure is that of Template:Em or Template:Em, defined as the join of all normal subgroups contained in $S.$

Group presentations

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