# Variety of groups is congruence-uniform

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-uniform variety. In other words, every group is a congruence-uniform algebra in the variety of groups. More explicitly, given any group and any congruence on it, all the congruence classes are of equal size.

## Translation to the language of groups

In the language of groups, the above statement interprets as: all the cosets of a normal subgroup have the same size.

## Related facts

### For analogous algebraic structures

Some similar algebraic structures for which the variety is congruence-uniform:

Some similar algebraic structures for which the variety is not congruence-uniform:

## Facts used

## Proof

### Proof idea

This follows from a more general fact for a group: the left cosets are in bijection via left multiplication, combined with the fact that for any congruence class on a group, the congruence classes are the cosets of a normal subgroup. This is essentially the statement of the first isomorphism theorem.