An extension, a section, and its failure
A [[group-extension|group extension]] of G by an abelian group A is a short exact sequence of groups 1 → A → E → G → 1. The conjugation of E on its normal subgroup A descends to a G-action on A (A is abelian, so inner automorphisms of A are trivial and the action is well defined). Fix a set-theoretic section s : G → E with s(1) = 1, picking one preimage of each g. There is no reason s respects multiplication; the defect lives in A.
Since s(g)s(h) and s(gh) have the same image g h in G, they differ by an element of A:
s(g) s(h) = f(g,h) * s(gh), f(g,h) in A.
This f : G x G -> A is the FACTOR SET (the 2-cochain). Associativity in E forces a relation.
Write the action as g.a = s(g) a s(g)^{-1}. Expand (s(g)s(h))s(k) = s(g)(s(h)s(k)):
g.f(h,k) - f(gh,k) + f(g,hk) - f(g,h) = 0.
That is EXACTLY d^2 f = 0: f is a 2-COCYCLE. (using additive notation in A)
Change the section: s'(g) = c(g) * s(g) with c: G -> A. Then
f'(g,h) - f(g,h) = g.c(h) - c(gh) + c(g) = (d^1 c)(g,h),
so f changes by a 2-COBOUNDARY. ==> the class [f] in H^2(G,A) is independent of the section.The classification theorem
This is the central fact of the track. Equivalence classes of extensions 1 → A → E → G → 1 inducing a fixed G-action on A are in bijection with [[second-cohomology|H^2(G, A)]]. The zero class corresponds to the split extension — the semidirect product A ⋊ G, where a homomorphic section exists, equivalently a split exact sequence. A nonzero class is a genuinely non-split extension that no choice of section can untwist.
When the action is trivial, A lands in the center of E and we speak of a [[central-extension|central extension]]; H^2(G, A) with trivial action classifies these. The classic specimen: with G = Z/2 acting trivially on A = Z/2, H^2 = Z/2, whose two classes split Klein four V from the cyclic group Z/4 — both are extensions of Z/2 by Z/2, distinguished precisely by their factor set.
G = Z/2 = {1, t}, A = Z/2 = {0,1}, trivial action. Sections s(1)=e, s(t)=x.
A factor set is determined by one value f(t,t) in A (others forced by s(1)=1).
f(t,t) = 0 : x^2 = s(t)s(t) = f(t,t)*s(t^2) = 0*e = e. E = Z/2 x Z/2 = V (split).
f(t,t) = 1 : x^2 = the nontrivial element of A. E = Z/4 (non-split).
Both f are 2-cocycles (trivial G means d^2 collapses to alternating-sum = 0 here),
and the only 2-coboundary is 0 because c: G->A with c(1)=0 gives d^1 c (t,t)= c(t)-c(1)+c(t)=2c(t)=0.
So H^2(Z/2, Z/2) = Z/2 <--> { V, Z/4 }. The two extension classes, seen directly.The Schur multiplier
Specialize to central extensions by C* (or by Q/Z). The group H^2(G, C*) with trivial action is the [[schur-multiplier|Schur multiplier]] M(G), historically the obstruction to lifting a projective representation of G to an honest linear one — a projective rep gives a factor set valued in C*, and it linearizes iff that class is zero in M(G). The Schur multiplier also governs the universal central extension of a perfect group and shows up later as K_2 in algebraic K-theory.