group extension
A group extension is a group assembled from two simpler pieces: a normal subgroup sitting inside, and the quotient by it sitting on top. Think of it as a two-story building where the ground floor is the normal subgroup N and the upper floor is the quotient Q; the building E contains both, but knowing the two floors does not by itself tell you how the staircase connects them. The connection data — the action and the twisting — is what makes extension theory rich.
Formally, an extension of a group Q by a group N is a short exact sequence 1 -> N -> E -> Q -> 1, meaning N is (isomorphic to) a normal subgroup of E and Q is (isomorphic to) the quotient E/N. Choosing a set-theoretic section s : Q -> E (with the quotient map undoing it) lets one transport the multiplication of E into data on N × Q: a conjugation action of Q on N (well defined up to inner automorphisms) and, when N is abelian, a factor set measuring the failure of s to be a homomorphism.
When N is abelian the extensions inducing a fixed action of Q on N are classified, up to equivalence, by the second cohomology group H^2(Q, N); the split extensions (those admitting a homomorphic section) form the zero class and are exactly the semidirect products. For nonabelian N the classification is more delicate: an action need not lift to an actual extension, and the obstruction lives in H^3(Q, Z(N)). Extension theory thus translates a structural question about groups into homological algebra.
1 -> Z/3Z -> S_3 -> Z/2Z -> 1 exhibits the symmetric group S_3 as an extension of Z/2Z by the cyclic group A_3 = Z/3Z. It is split (a transposition gives a homomorphic section), so S_3 is the semidirect product Z/3Z ⋊ Z/2Z.
S_3 as a split extension of Z/2Z by Z/3Z.
Two extensions are called equivalent when there is an isomorphism E -> E' that is the identity on N and induces the identity on Q. This is finer than mere isomorphism of E and E': distinct extension classes can have isomorphic middle groups.