International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 4, Pages 707-711
doi:10.1155/S0161171284000739
Admissible groups, symmetric factor sets, and simple algebras
Mathematics Department, University of Calgary, T2N 1N4, Canada
Received 22 June 1983
Copyright © 1984 R. A. Mollin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let K be a field of characteristic zero and suppose that D is a K-division algebra; i.e. a finite dimensional division algebra over K with center K. In Mollin [1] we proved that if K contains no non-trivial odd order roots of unity, then every finite odd order subgroup of D* the multiplicative group of D, is cyclic. The first main result of this paper is to generalize (and simplify the proof of) the above. Next we generalize and investigate the concept of admissible groups. Finally we provide necessary and sufficient conditions for a simple algebra, with an abelian maximal subfield, to be isomorphic to a tensor product of cyclic algebras. The latter is achieved via symmetric factor sets.