# Trivial pair of actions is compatible

## Statement

Suppose and are groups and and are group homomorphisms. Then, if and are both trivial, they form a compatible pair of actions.

## Related facts

- Compatible with trivial action iff image centralizes inner automorphisms
- Trivial pair of Lie ring actions is compatible

## Proof

We want to show that the trivial pair of actions is a compatible pair.

**Want to show**: If both the actions of the groups *on each other* are trivial, then the following holds, where is interpreted from context as the action , the action , or the action of a group on itself by conjugation:

**Proof**:

**First equality, left side**: where we use the triviality of the action of on .

**First equality, right side**: where the first step uses the triviality of the action of on and the second step uses the triviality of the action of on .

Thus, the first equality holds.

**Second equality, left side**: using the triviality of the action of on .

**Second equality, right side**: where the first step uses the triviality of the action of on and the second step uses the triviality of the action of on .