Subhomomorphism relation between transfer condition operator and composition operator

Statement

Let $T$ denote the Transfer condition operator (?) on subgroup properties: given a subgroup property $p$, the property $T(p)$ is the weakest property satisfying the transfer condition, that is stronger than $p$.

Let $*$ denote the Composition operator (?).

Then:

$T(p)*T(q)\leq T(p*q)$.

Related facts

Similar facts

• Subhomomorphism relation between transfer-closure operator and intermediately operator over composition operator

Corollaries

• Transfer condition is composition-closed
• Transitivity is transfer condition operator-preserved