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The Frattini Subgroup of GLn (D)

Ghasemi, Mahmood | 2013

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 45715 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Hazavei, Mohammad
  7. Abstract:
  8. Given a finite dimensional F-centeral simple algebra A = Mn(D), denote by A′ the derived group of its unit group A*. In this thesis, the Frattini subgroup Φ(A*) of A*for various fields F is investigated. Setting G = F* ∩ Φ(A*), when F is a local or global field the group G is completely determined. For global fields it is proved that when F is a real global field, then Φ(A*) = Φ(F*)Z(A′) otherwise Φ(A*) = ∩F*p where the intersection is taken over primes p not dividing the degree of A. Using the connection between Φ(A*) and Φ(F*) via Z(A′), Φ(A*) is also calculated for some particular division rings D.

  9. Keywords:
  10. Frattini Subgroup ; Multiplicative Group ; Local Field ; Global Field

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