Let D be a division algebra of degree m over its center F.
Herstein has shown that any finite normal subgroup of
D^{*}:=GL_{1}(D) is central. Here, as a generalization of this
result, it is shown that any finitely generated normal subgroup of
D^{*} is central. This also solves a problem raised by Akbari and
MahdaviHezavehi (Proc. Amer. Math. Soc., to appear) for
finitedimensional division algebras. The structure of maximal
multiplicative subgroups of an arbitrary division ring D is then
investigated. Given a maximal subgroup M of D^{*} whose center
is algebraic over F, it is proved that if M satisfies a
multilinear polynomial identity over F, then [D:F] < ∞.
Download TeX format
