Supersoluble crossed product criterion for division algebras

Crossed product conditions for division algebras of prime power degree

“…, g t are a linearly independent subset of G over K, and this completes the proof. P To prove our next lemma, we shall need the following results from [2].…”

“…A subgroup G of D * is said to be irreducible if F [G] = D. When n = p, a prime, it is shown in [5] that D is cyclic if and only if D * contains a nonabelian soluble subgroup. A criterion is also given in [2] for D to be a supersoluble (nilpotent) crossed product division algebra in terms of subgroups D * . More precisely, it is shown that D is a supersoluble (nilpotent) crossed product if and only if D * contains an abelian-by-supersoluble (abelian-by-nilpotent) irreducible subgroup.…”

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.…”