Variety of groups is congruence-permutable

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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-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: