On maximal subgroups of the multiplicative group of a division algebra

The question of existence of a maximal subgroup in the multiplicative group D∗ of a division algebra D finite-dimensional over its center F is investigated. We prove that if D∗ has no maximal subgroup, then deg(D) is not a power of 2, F∗2 is divisible, and for each odd prime p dividing deg(D), there exist noncyclic division algebras of degree p over F.