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.¹² 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.³

Suppose G is a group and A is an abelian group. For a group extension

${\displaystyle 1\to A\to X\to G\to 1,}$

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

${\displaystyle (g,a)*(h,b):=(gh,f(g,h)a^{\sigma (h)}b).}$

So f must be a "group 2-cocycle" (and thus define an element in H²(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 groups⁴

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.

Let G be a group and L a field on which G acts as automorphisms. A cocycle or (Noether) factor system^5: 31 is a map c: G × G → L^* satisfying

${\displaystyle c(h,k)^{g}c(hk,g)=c(h,kg)c(k,g).}$

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

${\displaystyle c'(g,h)=c(g,h)(a_{g}^{h}a_{h}a_{gh}^{-1}).}$

Cocycles of the form

${\displaystyle c(g,h)=a_{g}^{h}a_{h}a_{gh}^{-1}}$

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

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^5: 31 A, which is a K-algebra containing L as a subfield, generated by the elements λ in L and u_g with multiplication

${\displaystyle \lambda u_{g}=u_{g}\lambda ^{g},}$

${\displaystyle u_{g}u_{h}=u_{gh}c(g,h).}$

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

${\displaystyle A=(L,G,c).}$

The crossed product algebra A is a central simple algebra (CSA) of degree equal to [L : K].⁶ The converse holds: every central simple algebra over K that splits over L and such that deg A = [L : K] arises in this way.⁶ 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 H².⁷⁸

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

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

and then we have uⁿ = a in K. This element a specifies a cocycle c by^5: 33

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

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/KL^*. We obtain the isomorphisms

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