An extension of a group by a group is a group with a normal subgroup such that and . This information can be encoded into a short exact sequence of groups

where is injective and is surjective.

It should be noted that some authors reverse the roles and say that is an extension of (Spanier 1994, Mac Lane and Birkhoff 1993).

Given groups and there are (often) many extensions of by . Examples include the direct product of and and a semidirect product of and . A function such that is the identity function on is called a transversal function. 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.