# Variety of groups is congruence-permutable

From Groupprops

This article gives the statement, and possibly proof, of a property satisfied by the variety of groups

View a complete list of such property satisfactions

## Contents

## Statement

The variety of groups is a congruence-permutable variety. In other words, any two congruences on any group permute, or every group is a congruence-permutable algebra in the variety of groups.

## Translation to the language of groups

The translation of this statement to the ordinary language of groups is: given two normal subgroups of a group, their product is a normal subgroup. In other words, any two normal subgroups permute and their product is also a normal subgroup.

## Proof

### Direct proof

### Proof in terms of other results

The result can be thought of as a combination of two facts: