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.

Proof

This follows from a more general fact for a group: the left cosets are in bijection via left multiplication.