# 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 (x, y) 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 H$^2$(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 H$^2$(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 u$_g$ 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 H$^2$. We thus obtain an identification of the Brauer group, where the elements are classes of CSAs over K, with H$^2$.

### 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 = u$_t$ be the generator in A corresponding to t. We can define the other generators
    $u_{t^{i}}=u^{i}\,$
and then we have u$^n$ = a in K. This element a specifies a cocycle c by
    $c(t^{i},t^{j})={\begin{cases}1&{\text{if }}i+j<n,\\a&{\text{if }}i+j\geq n.\end{cases}}$
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$^*$/N$_{L/K}$L$^*$. We obtain the isomorphisms
    $\operatorname {Br} (L/K)\equiv K^{*}/N_{L/K}L^{*}\equiv H^{2}(G,L^{*}).$