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
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.
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:
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.