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Existence of nonabelian free subgroups in the maximal subgroups of GL n(D)

Dorbidi, H. R ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1142/S100538671400042X
  3. Abstract:
  4. Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of GL n(D) contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of Mn(D) such that NGLn<(D) (K*)=M, K* M, K/F is Galois with Gal(K/F) ≅ M/K*, and F[M]=Mn(D). In particular, when F is global or local, it is proved that if ([D:F],Char(F))=1, then every non-abelian maximal subgroup of GL 1(D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn(F) contains no solvable maximal subgroups provided that F is local or global and n ≥ 5. © 2014 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University
  5. Keywords:
  6. Central simple algebra ; Free subgroup ; Maximal subgroup
  7. Source: Algebra Colloquium ; Vol. 21, issue. 3 , 2014 , p. 483-496
  8. URL: http://www.worldscientific.com/doi/abs/10.1142/S100538671400042X