# Subnormality is finite-relative-intersection-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup) satisfying a subgroup metaproperty (i.e., finite-relative-intersection-closed subgroup property)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about subnormal subgroup |Get facts that use property satisfaction of subnormal subgroup | Get facts that use property satisfaction of subnormal subgroup|Get more facts about finite-relative-intersection-closed subgroup property

## Contents

## Statement

### Property-theoretic statement

The property of being a subnormal subgroup is finite-relative-intersection-closed.

### Statement with symbols

Suppose are subgroups of such that is a subnormal subgroup of , and is a subnormal subgroup inside some subgroup containing both and . Then, is subnormal in .

More specifically, if is -subnormal in and is -subnormal in , then is -subnormal in .

## Related facts

### Applications

- Subnormality is permuting join-closed: If and are both subnormal subgroups and they permute, i.e., , then is subnormal. The proof of this uses certain subgroups in intermediate steps that are subnormal as a consequence of the fact that subnormality is finite-relative-intersection-closed.

## Facts used

- Subnormality satisfies transfer condition: If is -subnormal and , then is -subnormal in .
- Subnormality is transitive: If are groups, and is -subnormal in and is -subnormal in , then is -subnormal in .# Transitive and transfer condition implies finite-relative-intersection-closed
- Transitive and transfer condition implies finite-relative-intersection-closed

## Proof

**Given**: . is -subnormal in and is -subnormal in a subgroup of containing both and .

**To prove**: is -subnormal in .

**Proof**:

- (
**Facts used**: fact (1);**Given data used**: , is -subnormal in ): is -subnormal in : is -subnormal in , and , so by fact (1), is -subnormal in . - (
**Facts used**: fact (2);**Given data used**: is -subnormal in ): is -subnormal in and is -subnormal in , so by fact (2), is -subnormal in .