Locally Finite Conditions on Maximal Subgroups of GL_n(D)

Abstract

Given a division ring D with center F, the structure of maximal subgroups M of GLn(D) is investigated. Suppose D ≠ F or n > 1. It is shown that if M/(M ∩ F*) is locally finite, then char F=p > 0 and either n=1, [D:F]=p2 and M ∪ {0} is a maximal subfield of D, or D=F, n=p, and M ∪ {0} is a maximal subfield of Mp(F), or D=F and F is locally finite. It is also proved that the same conclusion holds if M/(M ∩ F*) is torsion and D is of finite dimension over F. Furthermore, it is shown that if the r-th derived group M(r) of M is locally finite, then either M(r) is abelian or F is algebraic over its prime subfield.

ON SOME SUBGROUPS OF D*WHICH SATISFY A GENERALIZED GROUP IDENTITY

“…In [6, Theorem 2.10] (or see [8]), it is proved that if D * contains a maximal subgroup M such that M/(M ∩ F ) is locally finite, then D satisfies three properties in Remark 4.7. Here, a group G is called locally finite if every finitely generated subgroup of G is finite.…”