Normal closure (group theory)

Properties and description

Formally, if ${\displaystyle G}$ is a group and ${\displaystyle S}$ is a subset of ${\displaystyle G,}$ the normal closure ${\displaystyle {\mathrm{ncl}}_G(S)}$ of ${\displaystyle S}$ is the intersection of all normal subgroups of ${\displaystyle G}$ containing ${\displaystyle S}$: $${\displaystyle {\mathrm{ncl}}_G(S)=\bigcap _{S\subseteq N◃G}N.}$$

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

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

Therefore one can also write $${\displaystyle {\mathrm{ncl}}_G(S)=\{g_1^{-1}{s}_1^{{ϵ}_1}{g}_1\dots g_n^{-1}{s}_n^{{ϵ}_n}{g}_n:n\ge 0,{ϵ}_i=\pm 1,s_i\in S,g_i\in G\}.}$$

Any normal subgroup is equal to its normal closure. The conjugate 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 ${\displaystyle ⟨S^G⟩,}$ ${\displaystyle ⟨S{⟩}^G,}$ ${\displaystyle ⟨⟨S⟩{⟩}_G,}$ and ${\displaystyle ⟨⟨S⟩{⟩}^G.}$

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 ${\displaystyle S.}$