S. Akbari, M. Mahdavi-Hezavehi, M. G. Mahmudi: Maximal Subgroups of $GL_1(D)$

s_akbari@rose.ipm.ac.ir, mahdavi@rose.ipm.ac.ir, gmahmudi@rose.ipm.ac.ir

Submission: 1999, July 13

Let $D$ be a division ring of degree $m$ over its centre $F$. 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 in [1] for finite dimensional division rings. The structure of maximal multiplicative subgroups of an arbitrary division ring $D$ is then investigated. Given a maximal subgroup $M$ of $D^*$ whose centre is algebraic over $F$, it is proved that if $M$ satisfies a multilinear polynomial identity over $F$, then $[D : F] < \infty$.

1991 Mathematics Subject Classification: 15A33, 16K

Keywords and Phrases: Division rings, maximal subgroups, absolutely irreducible.

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