# Transitive and transfer condition implies finite-intersection-closed

## Contents

## 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 is termed transitive if whenever are groups such that satisfies property in and satisfies property in , also satisfies property in .

### Transfer condition

`Further information: Transfer condition`

A subgroup property is said to satisfy the transfer condition if whenever such that satisfies property in , satisfies property in .

### Finite-intersection-closed subgroup property

`Further information: Finite-intersection-closed subgroup property`

A subgroup property is termed finite-intersection-closed if whenever are subgroups satisfying property in , then also satisfies property in .

## Facts used

- Transitive and transfer condition implies finite-relative-intersection-closed
- 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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