Abelian coverings of finite general linear groups and an application to their non-commuting graphs
Azizollah Azad
, Mohammad A. Iranmanesh
, Cheryl E. Praeger
and Pablo Spiga
DOI: 10.1007/s10801-011-0288-2
Abstract
In this paper we introduce and study a family A n( q) \mathcal{A}_{n}(q) of abelian subgroups of GL n( q) {\rm GL}_{n}(q) covering every element of GL n( q) {\rm GL}_{n}(q). We show that A n( q) \mathcal{A}_{n}(q) contains all the centralizers of cyclic matrices and equality holds if q> n. For q>2, we obtain an infinite product expression for a probabilistic generating function for | A n( q) | |\mathcal{A}_{n}(q)|. This leads to upper and lower bounds which show in particular that
| c 1 q - n \sterling \frac | A n( q) | | GL n( q) | \sterling c 2 q - n c_1q^{-n}\leq \frac{|\mathcal{A}_n(q)|}{|\mathrm{GL}_n(q)|}\leq c_2q^{-n}
Pages: 683–710 Keywords: keywords general linear group; cyclic matrix; non-commuting subsets of finite groups; non-commuting graph Full Text: PDF References1. Abdollahi, A., Akbari, A., Maimani, H.R.: Non-commuting graph of a group. J. Algebra 298, 468-492 (2006)
2. Azad, A., Praeger, C.E.: Maximal sets of pairwise noncommuting elements of finite three-dimensional general linear groups. Bull. Aust. Math. Soc. 80, 91-104 (2009) 3. Bertram, E.A.: Some applications of graph theory to finite groups. Discrete Math. 44, 31-43 (1983) 4. Brown, R.: Minimal covers of Sn by abelian subgroups and maximal subsets of pairwise noncommuting elements. J. Comb. Theory, Ser. A 49(2), 294-307 (1988) 5. Carter, R.W.: Finite Groups of Lie Type, Conjugacy Classes and Complex Characters. Wiley, Chichester (1993) 6. Chin, A.M.Y.: On non-commuting sets in an extra special p-group. J. Group Theory 8, 189-194 (2005) 7. Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Pure and Applied Mathematics, vol. XI. Wiley, New York (1962) 8. Dixon, J.D.: Maximal abelian subgroups of the symmetric groups. Can. J. Math. 23, 426-438 (1971) 9. Fulman, J.: Cycle indices for the finite classical groups. J. Group Theory 2, 251-289 (1999) 10. Fulman, J., Neuman, P.N., Praeger, C.E.: A Generating Function Approach to the Enumeration of Matrices in Classical Groups over Finite Fields. Mem. AMS, vol. 176. AMS, Providence (2005) 11. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Clarendon Press, Oxford (1979) 12. Faber, V., Laver, R., Mckenzie, R.: Covering of groups by abelian subgroups. Can. J. Math. 30, 933- 945 (1978) 13. Neumann, B.H.: A problem of Paul Erd\Acute\Acute os on groups. J. Aust. Math. Soc. A 21, 467-472 (1976) 14. Neumann, P.M., Praeger, C.E.: Cyclic matrices over finite fields. J. Lond. Math. Soc. 52, 263-284 (1995) 15. Neumann, P.M., Praeger, C.E.: Cyclic matrices and the MEATAXE. In: Ohio State Univ. Math. Res. Inst. Publ., vol. 8, pp. 291-300. de Gruyter, Berlin (2001) 16. Pyber, L.: The number of pairwise non-commuting elements and the index of the centre in a finite group. J. Lond. Math. Soc. 35, 287-295 (1987) 17. Sloane, N.J.A. (ed.): The On-Line Encyclopedia of Integer Sequences. Published electronically at (2008) J Algebr Comb (2011) 34:683-710 18. Tomkinson, H.J.: Groups covered by finitely many cosets or subgroups. Commun. Algebra 15, 845- 859 (1987) 19. Vdovin, E.P.: The number of subgroups with trivial unipotent radicals in finite groups of Lie type.
© 1992–2009 Journal of Algebraic Combinatorics
|