# Group extension

This article describes a product notion for groups. See other related product notions for groups.

## Definition

Suppose and are (possibly isomorphic, possibly non-isomorphic) groups.

A **group extension** with normal subgroup and quotient group is defined as a group with a specified normal subgroup having a specified isomorphism to and a specified isomorphism from the quotient group to .

In some parts of group theory, such a is termed an extension *of* (the subgroup isomorphic to the normal subgroup) *by* (the subgroup isomorphic to the quotient group). In some other areas of mathematics, particularly geometric group theory and homology and cohomology theory, is termed an extension *of* the quotient *by* the normal subgroup, so in this case that would be an extension of by . A choice of terminology that avoids this confusion is "extension with normal subgroup and quotient group ."

A group extension with normal subgroup and quotient group can alternatively be thought of as a group along with a short exact sequence of groups:

The group extension problem seeks to classify all group extensions with a specified normal subgroup and a specified quotient group.

### Equivalence notion

There are various notions of equivalence for group extensions. The strongest notion is that of congruent group extension, where the specified isomorphisms *agree*. The equivalence classes of group extensions upto congruence, when nonempty, can be identified (though not canonically) with the second cohomology group .

A somewhat weaker notion is where the specified isomorphisms agree up to automorphisms in the groups and . Extensions which are equivalent under this weaker (broader) notion of equivalence are termed pseudo-congruent group extensions. The set of equivalence classes here is the set of interest from the perspective of the number of extensions with *distinct* behavior. This set can be viewed as a set of equivalence classes within the original set of congruence classes of group extensions. The new set could be very far from a group.