Subdistributivity relation between intersection operator and composition operator

Statement

Finite version

Suppose $p_{1},p_{2},q$ are subgroup properties. Let $\cap$ denote the Intersection operator (?) on subgroup properties, and $*$ denote the Composition operator (?) on subgroup properties. Then:

$(p_{1}\cap p_{2})*q\leq (p_{1}*q)\cap (p_{2}*q)$.

Related facts

Corollaries

• Finite-intersection-closedness is left residual-closed: This is a corollary of the finite version.
• Intersection-closedness is left residual-closed: This is a corollary of the infinite version.