Brauer-Clifford equivalence of full matrix algebras.
Let \(G\) be a finite group, \(F\) a field, and \(Z\) a commutative central simple \(G\)-algebra. In [J. Algebra 321, No. 12, 3620-3642 (2009; Zbl 1186.20011)], the author introduced the Brauer-Clifford group \(\text{BrCliff}(G,Z)\). This group consists of certain equivalence classes of central simple \(G\)-algebras, and the product is associated with the tensor product of algebras. In [op. cit.] it was proved that it has a natural homomorphism into a Brauer group, and that the kernel of this homomorphism is the set of all equivalence classes of \(G\)-algebras which are full matrix algebras. In the paper under review it is shown that this kernel is isomorphic to a certain second cohomology group. In certain situations it is also shown how to compute the element of the second cohomology group associated with a certain family of characters. The results in this paper are used in the author's proof of a strengthening of the McKay conjecture for \(p\)-solvable groups [in Adv. Math. 217, No. 5, 2170-2205 (2008; Zbl 1149.20010)].
scientific article; zbMATH DE number 5602623