Compatible with trivial action iff image centralizes inner automorphisms
Then, the following are equivalent:
- and form a compatible pair of actions.
- The image of in lies inside the centralizer , i.e., it commutes with all the inner automorphisms of .
The actions are compatible if and only if the following two conditions hold, where denotes the action or conjugation within a group, as is clear from context:
Using the triviality of , these conditions are equivalent to:
Note that the second condition is vacuously true, so it conveys no information. The first condition is equivalent to saying that the action of commutes with conjugation by , which is equivalent to saying that .