# Quotient group acts on abelian normal subgroup

From Groupprops

## Statement

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

given as follows:

where is conjugation by in .

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

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