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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
Mahdavi-Hezavehi ($Proc. Amer. Math. Soc.,$ to appear) for
finite-dimensional 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 $[D:F]<\infty$.
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