Group extension

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

Definition

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

A group extension with normal subgroup $A$ and quotient group $B$ is defined as a group $G$ with a specified normal subgroup $N$ having a specified isomorphism to $A$ and a specified isomorphism from the quotient group $G/N$ to $B$.

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

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

$1 \to A \to G \to B \to 1$

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 $H^2(B,Z(A))$.

A somewhat weaker notion is where the specified isomorphisms agree up to automorphisms in the groups $A$ and $B$. 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.