Submission: 2004, Mar 18
Let $D$ be a division ring with centre $ F $. Assume that $M$ is a maximal subgroup of $GL_n (D)$, $n\geq 1$ such that $ Z(M) $ is algebraic over $ F $. Group identities on $ M $ and polynomial identities on the $ F $-linear hull $ F[M] $ are investigated. It is shown that if $F[M]$ is a PI-algebra, then $[D : F] <\infty $. When $ D $ is noncommutative and $ F $ is infinite, it is also proved that if $ M $ satisfies a group identity and $ F[M] $ is algebraic over $ F $, then we have either $M=K^*$, where $K$ is a field and $[D:F]<\infty$ or $M$ is absolutely irreducible. For a finite dimensional division algebra $ D $, assume that $ N $ is a subnormal subgroup of $ GL_n(D) $ and $M$ is a maximal subgroup of $ N $. If $M$ satisfies a group identity, it is shown that $M$ is abelian-by-finite.
2000 Mathematics Subject Classification:
Keywords and Phrases:
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