Concept# Normal closure (group theory)

Summary

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

Official source

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