# Transiter master theorem for composition operator on subgroup properties

From Groupprops

## Statement

This is the version of the transiter master theorem for the composition operator on subgroup properties.

### The version for left transiters

Suppose is a subgroup property. Let denote the left transiter of .

- If is transitive, is stronger than its left transiter. That is, .
- If is identity-true, the left transiter of is stronger than . That is, .
- if is a t.i. subgroup property, i.e., is both transitive and identity-true, then .
- The left transiter of any property is a t.i. subgroup property, that is, it is both transitive and identity-true.

Together, (3) and (4) tell us that the left transiter operator is idempotent: .

### The version for right transiters

Suppose is a subgroup property. Let denote the right transiter of .

- If is transitive, is stronger than its right transiter. That is, .
- If is identity-true, the right transiter of is stronger than . That is, .
- if is a t.i. subgroup property, i.e., is both transitive and identity-true, then .
- The right transiter of any property is a t.i. subgroup property, that is, it is both transitive and identity-true.

Together, (3) and (4) tell us that the right transiter operator is idempotent: . Moreover, the fixed point space of this operator is the same as the fixed point space of the left transiter.