# Normal implies modular

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., normal subgroup) must also satisfy the second group property (i.e., modular subgroup)

View all group property implications | View all group property non-implications

Get more facts about normal subgroup|Get more facts about modular subgroup

## Statement

Suppose is a normal subgroup of . Then, for any subgroup of and any subgroup of such that , we have:

## Facts used

- Normal implies permutable
- Permutable implies modular (which in turn uses modular property of groups)

## Proof

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