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“School of Mathematics”

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Paper   IPM / M / 519
School of Mathematics
  Title:   Maximal subgroups of GL1(D)
  Author(s): 
1.  S. Akbari
2.  M. Mahdavi-Hezavehi
3.  M. G. Mahmudi
  Status:   Published
  Journal: J. Algebra
  No.:  2
  Vol.:  217
  Year:  1999
  Pages:   422-433
  Supported by:  IPM
  Abstract:
Let D be a division algebra of degree m over its center F. Herstein has shown that any finite normal subgroup of D*:=GL1(D) is central. Here, as a generalization of this result, it is shown that any finitely generated normal subgroup of D* is central. This also solves a problem raised by Akbari and Mahdavi-Hezavehi (Proc. Amer. Math. Soc., to appear) for finite-dimensional division algebras. The structure of maximal multiplicative subgroups of an arbitrary division ring D is then investigated. Given a maximal subgroup M of D* whose center is algebraic over F, it is proved that if M satisfies a multilinear polynomial identity over F, then [D:F] < ∞.

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