This group is for questions relating to "group extensions", a general means of describing a group in terms of a particular normal subgroup and quotient group.
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.
- A group extension is said to be split if there is a transversal function which is a homomorphism.
- A group extension is split iff it is a semidirect product.
- The study of group extensions has connections with group cohomology.
- The theory of group extensions is one of the cornerstones of homological algebra.
References:
https://en.wikipedia.org/wiki/Group_extension
https://www.encyclopediaofmath.org/index.php/Extension_of_a_group
