Automorphism group action lemma

Statement

Suppose $H$ is a group, and $N,G\leq H$ are subgroups such that $G\leq N_{H}(N)$. Suppose $\sigma '$ is an automorphism of $H$ such that the restriction of $\sigma '$ to $N$ gives an automorphism $\alpha$ of $N$, and such that $\sigma '$ also restricts to an automorphism of $G$, say $\sigma$. Consider the map:

$\rho :G\to \operatorname {Aut} (N)$

that sends an element $g\in G$ to the automorphism of $N$ induced by conjugation by $g$ (note that this is an automorphism since $G\leq N_{H}(N)$). Then, we have:

$\rho \circ \sigma =c_{\alpha }\circ \rho$

where $c_{\alpha }$ denotes conjugation by $\alpha$ in the group $\operatorname {Aut} (N)$.

Related facts

Similar facts

• Automorphism group action lemma for quotients

Applications

• Centerless and characteristic in automorphism group implies automorphism group is complete
• Semidirectly extensible implies linearly pushforwardable for representation over prime field