Quotient group acts on abelian normal subgroup

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

• Quotient group maps to outer automorphism group of normal subgroup
• Quotient by centralizer acts on normal subgroup