# Transiter master theorem for composition operator on subgroup properties

## Statement

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

### The version for left transiters

Suppose $p$ is a subgroup property. Let $L(p)$ denote the left transiter of $p$.

1. If $p$ is transitive, $p$ is stronger than its left transiter. That is, $p \le L(p)$.
2. If $p$ is identity-true, the left transiter of $p$ is stronger than $p$. That is, $L(p) \le p$.
3. if $p$ is a t.i. subgroup property, i.e., $p$ is both transitive and identity-true, then $p = L(p)$.
4. 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: $L(L(p)) = L(p)$.

### The version for right transiters

Suppose $p$ is a subgroup property. Let $R(p)$ denote the right transiter of $p$.

1. If $p$ is transitive, $p$ is stronger than its right transiter. That is, $p\le R(p)$.
2. If $p$ is identity-true, the right transiter of $p$ is stronger than $p$. That is, $R(p) \le p$.
3. if $p$ is a t.i. subgroup property, i.e., $p$ is both transitive and identity-true, then $p = R(p)$.
4. 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: $R(R(p)) = R(p)$. Moreover, the fixed point space of this operator is the same as the fixed point space of the left transiter.