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Maximal subgroups of GL n (D)

Akbari, S ; Sharif University of Technology | 2003

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  1. Type of Document: Article
  2. DOI: 10.1016/S0021-8693(02)00549-5
  3. Publisher: Academic Press Inc , 2003
  4. Abstract:
  5. 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
  6. Keywords:
  7. Division rings ; Irreducible ; Maximal subgroups ; Nilpotent ; Skew linear groups
  8. Source: Journal of Algebra ; Volume 259, Issue 1 , 2003 , Pages 201-225 ; 00218693 (ISSN)
  9. URL: https://www,sciencedirect.com/science/article/pii/S0021869302005495?via%3Dihub