In mathematics, a factor system (sometimes called factor set) is a fundamental tool of Otto Schreier’s classical theory for group extension problem.
In fact, A does not have to be abelian, but the situation is more complicated for non-abelian groups[4] If f is trivial, then X splits over A, so that X is the semidirect product of G with A.
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 ug with multiplication Equivalent factor systems correspond to a change of basis in A over K. We may write 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.
We thus obtain an identification of the Brauer group, where the elements are classes of CSAs over K, with H2.