Admissible Groups , Symmetric Factor Sets , and Simple Algebras

Let K be a field of characteristic zero and suppose that D is a K-fv.lslpn algebra; i.e. a finite dimensional division algebra over K with center K. In Mollin [i] 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 abellan maximal subfield, to be isomorphic to a tensor product of cyclic algebras. The latter is achieved via symmetric factor sets. Let K be a field of characteristic zero. We define the Schur subgroup $(K) of B(K), the Brauer group of K, to be those equivalence classes which ontaln a simple component of the group algebra KG for some finite group G. We let [A] denote the equivalence class of the K-central simple algebra A in B(K). The notation A B means [A] [B] in B(K). When A (R) B is written, the tensor product is assumed to be taken over the algebra in the left factor. For most basic results pertaining to S(K) the reader is referred to Yamada [2]. A crossed product algebra will be denoted (K/k,B) which is the central simple k-algebra having K-basis uo with o e G(K/k), subject to: uouT (o,T)uo and ux x u where x e K and o,T G(K/k), the Galois group of K over k. For further information pertaining to crossed products the reader is referred to Relner [3]. Finally we comment on notation. If m is a positive integer with m pa n where the prime p does not divide n then Iml p i.e. Iml denotes the p-part of m. A th P P primitive m root of unity will be denoted by m 2. SUBGROUPS OF SIMPLE ALGEBRAS. Let K be a field of characteristic zero. The major thrust of this section is to provide a generalization of Mollin [i, Theorem 3.6, p. 243]. To pave the road we first need a definition and some preliminary results.