Automorphism group action lemma

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Statement

Suppose H is a group, and N,G \le H are subgroups such that G \le 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 \le 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).

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