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
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