Transfer condition is composition-closed

This article gives the statement, and possibly proof, of a subgroup metaproperty (i.e., Transfer condition (?)) satisfying a subgroup metametaproperty (i.e., Composition-closed subgroup metaproperty (?))
View all subgroup metametaproperty satisfactions $|$ View all subgroup metametaproperty dissatisfactions

Statement

Property-theoretic statement

The subgroup metaproperty called the transfer condition satisfies the subgroup metametaproperty of being composition-closed.

Statement with symbols

Suppose $p,q$ are two subgroup properties satisfying the transfer condition. Then, the composition $p*q$ also satisfies the transfer condition.

Definitions used

Transfer condition

Further information: transfer condition

A subgroup property $p$ is said to satisfy the transfer condition if whenever $H\leq G$ has property $p$ in $G$, then for any subgroup $K\leq G$, $H\cap K$ has property $p$ in $K$.

Composition operator

Further information: composition operator

Given two subgroup properties $p,q$, the composition of $p$ and $q$, denoted $p*q$, is defined as follows. $H\leq G$ has property $p*q$ in $G$ if there exists an intermediate subgroup $K$ of $G$ such that $H$ has property $p$ in $K$ and $K$ has property $q$ in $G$.