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IPM
30
YEARS OLD

“School of Mathematics”

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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