Compatible with trivial action iff image centralizes inner automorphisms

Statement

Suppose $G$ and $H$ are groups with homomorphisms $\alpha :G\to \mathrm{A}\mathrm{u}\mathrm{t}(H)$ and $\beta :H\to \mathrm{A}\mathrm{u}\mathrm{t}(G)$ 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 $\mathrm{A}\mathrm{u}\mathrm{t}(G)$ lies inside the centralizer $C_{\mathrm{A}\mathrm{u}\mathrm{t}(G)}(\mathrm{I}\mathrm{n}\mathrm{n}(G))$, i.e., it commutes with all the inner automorphisms of $G$.

Related facts

• Trivial pair of actions is compatible

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:

$g_1\cdot (h\cdot g_2)=(g_1\cdot h)\cdot (g_1\cdot g_2)\mathrm{\forall }{g}_1,g_2\in G,h\in H$

$h_1\cdot (g\cdot h_2)=(h_1\cdot g)\cdot (h_1\cdot h_2)\mathrm{\forall }g\in G,h_1,h_2\in H$

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

$g_1\cdot (h\cdot g_2)=h\cdot (g_1\cdot g_2)\mathrm{\forall }{g}_1,g_2\in G,h\in H$

$h_1\cdot h_2=h_1\cdot h_2\mathrm{\forall }g\in G,h_1,h_2\in H$

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 $\beta (h)\in C_{\mathrm{A}\mathrm{u}\mathrm{t}(G)}(\mathrm{I}\mathrm{n}\mathrm{n}(G))$.