On normalizers of maximal subfields of division algebras

Document Type: Research Paper

Author

Faculty of Basic Sciences‎, ‎Babol Noshirvani University of Technology‎, ‎Shariati Ave.‎, ‎Babol‎, ‎Post Code 47148-71167‎, ‎Iran.

Abstract

‎Here‎, ‎we investigate a conjecture posed by Amiri and Ariannejad claiming‎ ‎that if every maximal subfield of a division ring $D$ has trivial normalizer‎, ‎then $D$ is commutative‎. ‎Using Amitsur classification of‎ ‎finite subgroups of division rings‎, ‎it is essentially shown that if‎ ‎$D$ is finite dimensional over its center then it contains a maximal‎ ‎subfield with non-trivial normalizer if and only if $D^*$ contains a‎ ‎non-abelian soluble subgroup‎. ‎This result generalizes a theorem of‎ ‎Mahdavi-Hezavehi and Tignol about cyclicity of division algebras of prime index.

Keywords

Main Subjects


M. Amiri and M. Ariannejad, Frobenius kernel and Wedderburn's little theorem, Bull. Iranian Math. Soc. 40 (2014), no. 4, 961--965.

S.A. Amitsur, Finite subgroups of division rings, Trans. Amer. Math. Soc. 80 (1955) 361--386.

P.K. Draxl, Skew Fields, Cambridge Univ. Press, Cambridge, 1983.

R. Hazrat, M. Mahdavi-Hezavehi and M. Motiee, Multiplicative groups of division rings, Math. Proc. R. Ir. Acad. 114A (2014), no. 1, 37--114.

R. Hazrat and A.R. Wadsworth, On maximal subgroups of the multiplicative group of a division algebra, J. Algebra 322 (2009), no. 7, 2528--2543.

T. Keshavarzipour and M. Mahdavi-Hezavehi, Crossed product conditions for central simple algebras in terms of irreducible subgroups, J. Algebra 315 (2007), no. 2, 738--744.

T.Y. Lam, A First Course in Noncommutative Rings, Grad. Texts in Math. 131, Springer-Verlag, 2nd edition, New York, 2001.

M. Mahdavi-Hezavehi, Free subgroups in maximal subgroups of GL1(D), J. Algebra 241 (2001), no. 2, 720--730.

M. Mahdavi-Hezavehi and M. Motiee, Division algebras with radicable multiplicative groups, Comm. Algebra 39 (2011), no. 11, 4084--4096.

M. Mahdavi-Hezavehi and M. Motiee, A criterion for the triviality of G(D) and its applications to the multiplicative structure of D, Comm. Algebra 40 (2012), no. 7, 2645--2670.

M. Mahdavi-Hezavehi and J.-P. Tignol, Cyclicity conditions for division algebras of prime degree, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3673--3676.

D.J.S. Robinson, A Course in The Theory of Groups, Grad. Texts in Math. 80, Springer-Verlag, 2nd edition, New York, 1996.

J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972) 250--270.

J.H.M. Wedderburn, A theorem on finite algebras, Trans. Amer. Math. Soc. 6 (1905), no. 3, 349--352.


Volume 43, Issue 6
November and December 2017
Pages 2051-2056
  • Receive Date: 18 September 2015
  • Revise Date: 21 December 2016
  • Accept Date: 22 December 2016