# Compatible with trivial action iff image centralizes inner automorphisms

## 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 $G$.

## 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 $\beta(h) \in C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G))$.