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“School of Mathematics”

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Paper   IPM / M / 767
School of Mathematics
  Title:   Maximal subgroups of GLn(D)*
  Author(s): 
1.  S. Akbari
2.  R. Ebrahimian
3.  H. Momenaee Kermani
4.  A. Salehi Golsefidy
  Status:   Published
  Journal: J. Algebra
  Vol.:  259
  Year:  2003
  Pages:   201-225
  Supported by:  IPM
  Abstract:
In this paper we study the structure of locally solvable, solvable, locally nilpotent and nilpotent maximal subgroups of skew linear groups. In [3] it has been conjectured that if D is a division ring and M a nilpotent maximal subgroup of D*, then D is commutative. In relation to this conjecture we show that if F[M]\F contains at least an algebraic element over F, then M is an abelian group. Also we show that \mathbbC* ∪\mathbbC* j is a solvable maximal subgroup of real quaternions and so give a counterexample to Conjecture 3 of [3], which states 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 nonabelian algebraic locally solvable maximal subgroup, which gives a full solution to both cases given in Theorem 8 of [3]. Ultimately, we extend our results to the general skew linear group.

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