Crossed product conditions for division algebras of prime power degree

Abstract

Let D be an F -central division algebra of degree p r , p a prime. A set of criteria is given for D to be a crossed product in terms of irreducible soluble or abelian-by-finite subgroups of the multiplicative group D * of D. Using the Amitsur's classification of finite subgroups of D * and the Tits alternative, it is shown that D is a crossed product if and only if D * contains an irreducible soluble subgroup. Further criteria are also presented in terms of irreducible abelian-by-finite subgroups and irreducible subgroups satisfying a group identity. Using the above results, it is shown that if D * contains an irreducible finite subgroup, then D is a crossed product.

Crossed product conditions for central simple algebras in terms of irreducible subgroups

“…We remark that the last result generalizes the case where the degree of D is a prime power, which is dealt with in [4]. Let G be an irreducible soluble subgroup of D * .…”

“…In [4] and [10], the authors ask whether a division algebra generated by a soluble-by-finite irreducible subgroup G is necessarily a crossed product over a maximal subfield. For some special cases where D is of a prime power degree and G is soluble or finite, the answer to the above mentioned question is shown to be positive.…”

Multiplicative groups of division rings

“…In [13], it was proved that if G is soluble, then D is a quasi-crossed product division algebra in the sense that it contains a tower of subfields F = Z(D) K ⊆ L such that K/F is Galois, L is a maximal subfield and L/K is an abelian Galois extension. Also, the results of [14] and [47] assert that D is a crossed product division algebra if one of the following conditions holds:…”

“…The first significant result concerning Question 2.5, was obtained in [60] where the author showed that if G is a soluble maximal subgroup (this means that G is a maximal subgroup of D * that is soluble), then the question has a positive answer. Afterwards, this question became the subject of a series of papers including [13], [14], [47] and [45]. In [13], it was proved that if G is soluble, then D is a quasi-crossed product division algebra in the sense that it contains a tower of subfields F = Z(D) K ⊆ L such that K/F is Galois, L is a maximal subfield and L/K is an abelian Galois extension.…”