Factor system

In mathematics, a factor system (sometimes called factor set) is a fundamental tool of Otto Schreier’s classical theory for group extension problem. It consists of a set of automorphisms and a binary function on a group satisfying certain condition (so-called cocycle condition). In fact, a factor system constitutes a realisation of the cocycles in the second cohomology group in group cohomology.

Introduction
Suppose $G$ is a group and $A$ is an abelian group. For a group extension
 * $$1 \to A \to X \to G \to 1,$$

there exists a factor system which consists of a function $f : G × G → A$ and homomorphism $σ: G → Aut(A)$ such that it makes the cartesian product $G × A$ a group $X$ as
 * $$(g,a)*(h,b) := (gh, f(g,h)a^{\sigma(h)}b).$$

So $f$ must be a "group 2-cocycle" (and thus define an element in H$2$(G, A), as studied in group cohomology). In fact, $A$ does not have to be abelian, but the situation is more complicated for non-abelian groups

If $f$ is trivial, then $X$ splits over $A$, so that $X$ is the semidirect product of $G$ with $A$.

If a group algebra is given, then a factor system f modifies that algebra to a skew-group algebra by modifying the group operation $xy$ to $f&thinsp;(x, y)&hairsp;xy$.

Application: for Abelian field extensions
Let G be a group and L a field on which G acts as automorphisms. A cocycle or (Noether) factor system  is a map c: G × G → L* satisfying


 * $$c(h,k)^g c(hk,g) = c(h,kg) c(k,g) . $$

Cocycles are equivalent if there exists some system of elements a : G → L* with


 * $$c'(g,h) = c(g,h) (a_g^h a_h a_{gh}^{-1}) . $$

Cocycles of the form


 * $$c(g,h) = a_g^h a_h a_{gh}^{-1} $$

are called split. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group H2(G,L*).

Crossed product algebras
Let us take the case that G is the Galois group of a field extension L/K. A factor system c in H2(G,L*) gives rise to a crossed product algebra A, which is a K-algebra containing L as a subfield, generated by the elements λ in L and ug with multiplication


 * $$\lambda u_g = u_g \lambda^g ,$$
 * $$u_g u_h = u_{gh} c(g,h) .$$

Equivalent factor systems correspond to a change of basis in A over K. We may write


 * $$A = (L,G,c) .$$

The crossed product algebra A is a central simple algebra (CSA) of degree equal to [L : K]. The converse holds: every central simple algebra over K that splits over L and such that deg A = [L : K] arises in this way. The tensor product of algebras corresponds to multiplication of the corresponding elements in H2. We thus obtain an identification of the Brauer group, where the elements are classes of CSAs over K, with H2.

Cyclic algebra
Let us further restrict to the case that L/K is cyclic with Galois group G of order n generated by t. Let A be a crossed product (L,G,c) with factor set c. Let u = ut be the generator in A corresponding to t. We can define the other generators


 * $$ u_{t^i} = u^i \, $$

and then we have un = a in K. This element a specifies a cocycle c by


 * $$c(t^i,t^j) = \begin{cases} 1 & \text{if } i+j < n, \\ a & \text{if } i+j \ge n. \end{cases} $$

It thus makes sense to denote A simply by (L,t,a). However a is not uniquely specified by A since we can multiply u by any element λ of L* and then a is multiplied by the product of the conjugates of λ. Hence A corresponds to an element of the norm residue group K*/NL/KL*. We obtain the isomorphisms


 * $$\operatorname{Br}(L/K) \equiv K^*/N_{L/K} L^* \equiv H^2(G,L^*) . $$