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

## Contents

## Statement

Suppose is a Transitive subgroup property (?) satisfying the Transfer condition (?). Then, is a Finite-relative-intersection-closed subgroup property (?).

## Definitions used

### Transitive subgroup property

`Further information: Transitive subgroup property`

A subgroup property is termed **transitive** if whenever are groups such that has property in , and has property in , we have that has property in .

### Transfer condition

`Further information: Transfer condition`

A subgroup property is said to satisfy the **transfer condition** if whenever are subgroups such that satisfies property in , we have that satisfies property in .

### Finite-relative-intersection-closed subgroup property

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

A subgroup property is termed **finite-relative-intersection-closed** if whenever are subgroups such that satisfies property in and satisfies property in some subgroup containing both and , then satisfies property in .

## Examples

- Subnormal subgroup: Subnormality is finite-relative-intersection-closed follows from the facts that subnormality is transitive and subnormality satisfies transfer condition.

## Proof

**Given**: A group with subgroups such that satisfies property in and satisfies property in some subgroup containing both and . is both transitive and satisfies the transfer condition.

**To prove**: satisfies property in .

**Proof**:

- (
**Given data used**: satisfies in , satisfies transfer condition): satisfies property in : satisfies property in , and is a subgroup of . Since satisfies the transfer condition, we have that satisfies property in . - (
**Given data used**: is transitive, satisfies in ): By the previous step, we have satisfies property in , and satisfies property in . Since is transitive, we obtain that satisfies in .