Crossed product conditions for division algebras of prime power degree. (Q1770498)

Let \(F\) be a field and \(D\) a central division \(F\)-algebra of degree \(n\). A subgroup \(G\) of the multiplicative group \(D^*\) of \(D\) is said to be irreducible, if \(F[G]=D\). Under the hypothesis that \(n=p^r\), where \(p\) is prime and \(r\) is a positive integer, the paper under review gives several criteria for \(D\) to be a crossed product. The authors show that \(D\) has this property if and only if \(D^*\) possesses an irreducible solvable subgroup. They prove that \(D\) is a crossed product, in case \(D^*\) has an irreducible finite subgroup. When \(p>2\) or \(\text{char}(F)=0\), it is established that \(D\) is such a product if and only if \(D^*\) contains an irreducible subgroup satisfying a group identity. The proof of the third result relies on the Tits alternative, namely the fact that a finitely generated linear group contains a noncyclic free subgroup, unless it is solvable-by-finite (see [\textit{J. Tits}, J. Algebra 20, 250-270 (1972; Zbl 0236.20032)]).