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

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:

• Variety of loops is congruence-uniform
• Variety of Lie rings is congruence-uniform

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

• Variety of monoids is not congruence-uniform

Facts used

1. Left cosets are in bijection via left multiplication
2. First isomorphism theorem

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.