Quotient group acts on abelian normal subgroup
From Groupprops
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:
given as follows:
where cg 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.
