# Modularity satisfies intermediate subgroup condition

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., modular subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)

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 modular subgroup |Get facts that use property satisfaction of modular subgroup | Get facts that use property satisfaction of modular subgroup|Get more facts about intermediate subgroup condition

## Contents

## Statement

### Verbal statement

A modular subgroup of a group is also a modular subgroup inside every intermediate subgroup.

## Definitions used

### Modular subgroup

`Further information: Modular subgroup`

A subgroup of a group is termed **modular** in if for any subgroups such that , we have:

.

### Intermediate subgroup condition

`Further information: Intermediate subgroup condition`

A subgroup property is said to satisfy the intermediate subgroup condition if whenever are groups such that satisfies property in , also satisfies property in .

## Related facts

### Related subgroup properties satisfying intermediate subgroup condition

- Permutability satisfies intermediate subgroup condition
- Normality satisfies intermediate subgroup condition
- Ellipticity satisfies intermediate subgroup condition

## Proof

**Given**: A group , subgroups . is modular in .

**To prove**: is modular in : whenever are such that , we have:

.

**Proof**: Since are subgroups of , they are also subgroups of , and the condition holds by assumption. Thus, by modularity of in , we conclude that:

.