# Compatible pair of actions

## Contents

## Definition

### Definition with left action convention

Suppose and are groups. Suppose is a homomorphism of groups, defining a group action of on . Suppose is a homomorphism of groups, defining a group action of on . For , denote by the conjugation map by . See group acts as automorphisms by conjugation. Then, we say that the actions form a **compatible pair** if both these conditions hold:

The above expressions are easier to write down if we use to denote all the actions. In that case, the conditions read:

Here is an equivalent formulation of these two conditions that is more convenient:

- (the here is the of the preceding formulation).
- (the here is the of the preceding formulation)

In the notation, these become:

- (the here is the of the preceding formulation).
- (the here is the of the preceding formulation)

### Definition with right action convention

We can give a corresponding definition using the right action convention, but the literature uses the left action convention, so this definition is intended purely as an illustrative exercise.**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

## Symmetry in the definition

Given groups and with actions and , is compatible with if and only if is compatible with . In other words, the definition of compatibility is symmetric under interchanging the roles of the two groups.

## Particular cases

- Trivial pair of actions is compatible: If both the actions are trivial, i.e., both the homomorphisms are trivial maps, then they form a compatible pair.
- Compatible with trivial action iff image centralizes inner automorphisms: Suppose the homomorphism is trivial. In that case, the homomorphism is compatible with if and only if the image of in is in the centralizer .
- Conjugation actions between subgroups that normalize each other are compatible: If are both subgroups of some group that normalize each other (i.e., each is contained in the normalizer of the other), and are the actions of the groups on each other by conjugation, then they form a compatible pair. Note that in this case, all the actions are just conjugation in and checking the conditions simply amounts to checking two words to be equal.