Transitive and transfer condition implies finite-intersection-closed

Statement

A Transitive subgroup property (?) that satisfies the Transfer condition (?) is finite-intersection-closed.

Definitions used

Transitive subgroup property

Further information: Transitive subgroup property

A subgroup property $p$ is termed transitive if whenever $H\leq K\leq G$ are groups such that $H$ satisfies property $p$ in $K$ and $K$ satisfies property $p$ in $G$, $H$ also satisfies property $p$ in $G$.

Transfer condition

Further information: Transfer condition

A subgroup property $p$ is said to satisfy the transfer condition if whenever $H,K\leq G$ such that $H$ satisfies property $p$ in $G$, $H\cap K$ satisfies property $p$ in $K$.

Finite-intersection-closed subgroup property

Further information: Finite-intersection-closed subgroup property

A subgroup property $p$ is termed finite-intersection-closed if whenever $H,K$ are subgroups satisfying property $p$ in $G$, then $H\cap K$ also satisfies property $p$ in $G$.

Facts used

1. Transitive and transfer condition implies finite-relative-intersection-closed
2. Finite-relative-intersection-closed implies finite-intersection-closed

Proof

Proof using given facts

The proof follows directly by piecing together facts (1) and (2).

Hands-on proof

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