Maximal subgroups of GLn(D)

In this paper we study the structure of locally solvable, solvable, locally nilpotent, and nilpotent maximal subgroups of skew linear groups. In [S. Akbari et al., J. Algebra 217 (1999) 422–433] it has been conjectured that if D is a division ring and M a nilpotent maximal subgroup of D∗, then D is commutative. In connection with this conjecture we show that if F [M]\F contains an algebraic element over F , then M is an abelian group. Also we show that C∗ ∪ C∗j is a solvable maximal subgroup of real quaternions and so give a counterexample to Conjecture 3 of [S. Akbari et al., J. Algebra 217 (1999) 422–433], which states that if D is a division ring and M a solvable maximal subgroup of D∗, then D is commutative. Also we completely determine the structure of division rings with a non-abelian algebraic locally solvable maximal subgroup, which gives a full solution to both cases given in Theorem 8 of [S. Akbari et al., J. Algebra 217 (1999) 422–433]. Ultimately, we extend our results to the general skew linear groups.  2002 Elsevier Science (USA). All rights reserved.