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

Properties and description

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

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

The subgroup ncl G ⁡ ( S ) {\displaystyle \operatorname {ncl} _{G}(S)} is the subgroup generated by the set S G = { s g : s ∈ S , g ∈ G } = { g − 1 s g : s ∈ S , g ∈ 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 S {\displaystyle S} in G . {\displaystyle G.} Therefore, one can also write the subgroup as the set of all products of conjugates of elements of S {\displaystyle S} or their inverses: ncl G ⁡ ( S ) = { g 1 − 1 s 1 ϵ 1 g 1 ⋯ g n − 1 s n ϵ n g n : n ≥ 0 , ϵ i = ± 1 , s i ∈ S , g i ∈ 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 } is the trivial subgroup.

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

Group presentations

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