# 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 $H \le K \le G$ 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 $H, K \le G$ 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 $G$.

## Proof

### Proof using given facts

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