D. Kiani, M. Mahdavi-Hezavehi: Identities on Maximal Subgroups of $GL_n(D)$

kiani@mehr.sharif.edu, mahdavih@sharif.edu

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:

Full text: dvi.gz 20 k, dvi 50 k, ps.gz 490 k, pdf.gz 115 k, pdf 132 k.