Compatible with trivial action iff image centralizes inner automorphisms

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Statement

Suppose G and H are groups with homomorphisms \alpha:G \to \operatorname{Aut}(H) and \beta:H \to \operatorname{Aut}(G) such that \alpha is the trivial homomorphism, i.e., it sends every element of G to the identity automorphism of H.

Then, the following are equivalent:

  1. \alpha and \beta form a compatible pair of actions.
  2. The image of \beta in \operatorname{Aut}(G) lies inside the centralizer C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G)), i.e., it commutes with all the inner automorphisms of G.

Related facts

Proof

The actions are compatible if and only if the following two conditions hold, where \cdot denotes the action \alpha,\beta or conjugation within a group, as is clear from context:

g_1 \cdot (h \cdot g_2) = (g_1 \cdot h) \cdot (g_1 \cdot g_2) \ \forall \ g_1,g_2 \in G, h \in H

h_1 \cdot (g \cdot h_2) = (h_1 \cdot g) \cdot (h_1 \cdot h_2) \ \forall g \in G, h_1,h_2 \in H

Using the triviality of \alpha, these conditions are equivalent to:

g_1 \cdot (h \cdot g_2) = h \cdot (g_1 \cdot g_2) \ \forall \ g_1,g_2 \in G, h \in H

h_1 \cdot h_2 = h_1 \cdot h_2 \ \forall g \in G, h_1,h_2 \in H

Note that the second condition is vacuously true, so it conveys no information. The first condition is equivalent to saying that the action of h commutes with conjugation by g_1, which is equivalent to saying that \beta(h) \in C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G)).