Quotient group acts on abelian normal subgroup

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Statement

Suppose G is a group and N is an Abelian normal subgroup. Then, the quotient group G/N has a naturally induced action on N, i.e., there is a homomorphism:

G/N \to \operatorname{Aut}(N)

given as follows:

gN \mapsto c_g

where c_g is conjugation by g in G.

The action is faithful if and only if N is also a self-centralizing subgroup.

We need N to be normal for the conjugation action to define an automorphism of N, and we need N to be Abelian for the map to be well-defined and independent of the choice of the coset representative.

Generalizations