factor set

A factor set is the precise instruction manual for multiplying inside a group extension. When you build a group E from a normal subgroup A and a quotient Q, you pick one representative in E for each element of Q; but the product of two representatives may not be the representative of the product — it overshoots by an element of A. The factor set records all these discrepancies, and the requirement that multiplication in E be associative becomes a clean equation on the factor set.

Given an extension 1 -> A -> E -> Q -> 1 with A abelian and a chosen set-section s : Q -> E, the factor set is the function f : Q × Q -> A defined by s(g)s(h) = f(g, h) · s(gh). Associativity of E forces the 2-cocycle identity g·f(h, k) - f(gh, k) + f(g, hk) - f(g, h) = 0. Choosing a different section changes f by a 2-coboundary, so the class [f] in H^2(Q, A) is an invariant of the extension. Conversely every 2-cocycle defines a group law on A × Q realizing that class.

Factor sets give the most hands-on description of H^2: an element of H^2(Q, A) literally is a factor set up to coboundary. The same data appears in the theory of central simple algebras as crossed-product algebras, where a factor set on a Galois group assembles a division-algebra structure, and the cohomology class is the algebra's class in the Brauer group. Normalizing f(1, g) = f(g, 1) = 0 corresponds to choosing s(1) = 1 and is always possible.