A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

REFERENCES

• Akbari , S. , Ebrahimian , R. , Momenaee Kermani , H. , Salehi Golsefidy , A. ( 2003 ). Maximal subgroups of GL n (D) . J. Algebra 259 : 201 – 225 .
   Web of Science ®Google Scholar
• Akbari , S. , Mahdavi-Hezavehi , M. , Mahmudi , M. G. ( 1999 ). Maximal subgroups of GL 1(D) . J. Algebra 217 : 422 – 433 .
   Web of Science ®Google Scholar
• Akbari , S. , Mahdavi-Hezavehi , M. (2002). On the existence of normal maximal subgroups in division rings. J. Pure Appl. Algebra 171:123–131.
   Web of Science ®Google Scholar
• Draxl , P. K. ( 1983 ). Skew Fields. London Math. Soc. Lecture Notes Ser . Vol. 81 . Cambridge : Cambridge Univ. Press .
   Google Scholar
• Ershov , Yu. ( 1983 ). Henselian valuations of division rings and the group SK 1 . Math USSR-Sb. 45 : 63 – 71 .
   Google Scholar
• Feit , W. , Thompson , J. G. ( 1963 ). Solvability of groups of odd order . Pacific J. Math. 13 : 775 – 1029 .
   Web of Science ®Google Scholar
• Hazrat , R. ( 2001 ). SK 1-like functors for division algebras . J. Algebra 239 : 573 – 588 .
   Web of Science ®Google Scholar
• Hazrat , R. ( 2002 ). Wedderburn's factorization theorem, Application to reduced K-theory . Proc. Amer. Math. Soc. 130 ( 2 ): 311 – 314 .
   Google Scholar
• Hazrat , R. ( 2008 ). On the first Galois cohomology group of the algebraic group SL1(D) . Comm. Algebra 36 ( 2 ): 381 – 387 .
   Web of Science ®Google Scholar
• Hazrat , R. , Mahdavi-Hezavehi , M. , Mirzaii , B. ( 1999 ). Reduced K-theory and the Group G(D) = D*/F*D′ . Algebraic K-theory and its applications. (Trieste, 1997) , River Edge , NJ : World Sci. Publ. , pp. 403–409.
   Google Scholar
• Hazrat , R. , Vishne , U. ( 2005 ). Triviality of the functor Coker (K 1(F) → K 1(D)) for division algebras . Comm. Algebra 33 : 354 – 361 .
   Google Scholar
• Hazrat , R. , Wadsworth , A. R. ( 2007 ). Nontriviality of certain quotients of K 1 groups of division algebras . J. Algebra 312 : 354 – 361 .
   Google Scholar
• Hazrat , R. , Wadsworth , A. R. ( 2009 ). On maximal subgroups of the multiplicative group of a division algebra . J. Algebra 322 : 2528 – 2543 .
   Web of Science ®Google Scholar
• Jacob , B. , Wadsworth , A. ( 1990 ). Division algebras over Henselian fields . J. Algebra 128 : 126 – 179 .
   Web of Science ®Google Scholar
• Keshavarzipuor , T. , Mahdavih-Hezavehi , M. ( 2005 ). On the non-triviality of G(D) and the existence of maximal subgrups of GL 1(D) . J. Algebra 285 : 213 – 221 .
   Google Scholar
• Keshavarzipour , T. , Mahdavi-Hezavehi , M. ( 2007 ). Crossed product conditions for central simple algebras in terms of irreducible subgroups. J. Algebra 315(2): 738–744 .
   Google Scholar
• Kiani , D. , Ramezan-nassab , M. ( 2009 ). Crossed product conditions for central simple algebras in terms of splitting fields . International Mathematical Forum 4 ( 32 ): 1587 – 1590 .
   Google Scholar
• Lam , T. Y. ( 2001 ). A First Course in Noncommutative Rings . New York : Springer-Verlag .
   Google Scholar
• Mahdavi-Hezavehi , M. ( 1998 ). Commutator subgroups of finite dimensional division algebras. Rev. Roumaine math. Pures Appl. 43:9–10, 853–867 .
   Google Scholar
• Mahdavi-Hezavehi , M. ( 2001 ). Free subgroups in maximal subgroups of GL 1(D) . J. Algebra. 241 : 720 – 730 .
   Web of Science ®Google Scholar
• Mahdavi-Hezavehi , M. ( 2002 ). Maximal subgroups of skew linear groups . Algebra Colloq. 9 : 1 – 6 .
   Google Scholar
• Mahdavi-Hezavehi , M. , Motiee , M. ( 2011 ). Division algebras with radicable multiplicative groups . Comm. Algebra 39 : 4084 – 4096 .
   Web of Science ®Google Scholar
• Mahdavi-Hezavehi , M. , Tignol , J.-P. ( 2003 ). Cyclicity conditions for division algebras of prime degree . Proc. Amer. Math. Soc. 131 ( 12 ): 3673 – 3676 .
   Web of Science ®Google Scholar
• Merkurjev , A. S. ( 1983 ). Brauer groups of fields . Comm. Algebra 11 : 2611 – 2624 .
   Web of Science ®Google Scholar
• Pierce , R. S. ( 1996 ). Associative Algebras . New York : Springer-Verlag .
   Google Scholar
• Rapinchuk , A. S. , Segev , Y. , Seitz , G. M. ( 2002 ). Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable . J. Amer. Math. Soc. 15 : 929 – 978 .
   Web of Science ®Google Scholar
• Robinson , D. J. (1982). A Course in the Theory of Groups. Grad. Text in Math., Vol. 80. Springer-Verlag.
   Google Scholar
• Robinson , D. J. ( 1972 ). Finiteness Conditions and Generalized Soluble Groups . Part 2 , Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 63. New York, Berlin : Springer-Verlag .
   Google Scholar
• Tignol , J.-P. , Wadsworth , A. R. ( 1987 ). Totally ramified valuations on finite-dimensional division algebras . Trans. Amer. Math. Soc. 302 ( 1 ): 223 – 250 .
   Web of Science ®Google Scholar
• Wadsworth , A. R. ( 1986 ). Extending valuations to finite dimensional division algebras . Proc. Amer. Math. Soc. 98 : 20 – 22 .
   Web of Science ®Google Scholar
• Communicated by A. Wadsworth.
   Google Scholar