Sylow multiplicities in finite groups

Document Type : Ischia Group Theory 2016

Author

Italy

Abstract

Let $G$ be a finite group and let $\mathcal{P}=P_{1},\ldots,P_{m}$ be a sequence‎ ‎of Sylow $p_{i}$-subgroups of $G$‎, ‎where $p_{1},\ldots,p_{m}$ are the distinct‎ ‎prime divisors of $\left\vert G\right\vert $‎. ‎The Sylow multiplicity of $g\in‎ ‎G$ in $\mathcal{P}$ is the number of distinct factorizations $g=g_{1}\cdots‎ ‎g_{m}$ such that $g_{i}\in P_{i}$‎. ‎We review properties of the solvable‎ ‎radical and the solvable residual of $G$ which are formulated in terms of‎ ‎Sylow multiplicities‎, ‎and discuss some related open questions‎.

Keywords

Main Subjects

References

[1] M. J. J. Barry and M. B. Ward, Products of Sylow Groups, Arch. Math., 63 (1994) 289–290.
[2] P. X. Gallagher, Group characters and Sylow subgroups, J. London Math. Soc., 39 (1964) 720–722.
[3] The GAP Group, GAP–Groups, Algorithms, and Programming, Version 4.4.12 of 2008, www.gap-system.org.
[4] P. Hall, A characteristic property of soluble groups, J. London Math. Soc., 12 (1937) 198–200.
[5] W. Herfort and D. Levy, Prosolvability criteria and properties of the prosolvable radical via Sylow sequences, Journal of Group theory, 19 (2016) 435–453.
[6] D. F. Holt and P. Rowley, On products of Sylow subgroups in finite groups, Arch. Math., 60 (1993) 105–107.
[7] B. Huppert, Character Theory of Finite Groups, Walter de Gruyter & Co, 1997.
[8] I. M. Isaacs, Character Theory of Finite Groups, Dover Publications, 1994.
[9] G. Kaplan and D. Levy, Sylow Products and the Solvable Radical, Arch. Math., 85 (2005) 304–312.
[10] G. Kaplan and D. Levy, The Solvable Radical of Sylow Factorizable Groups, Arch. Math., 85 (2005) 490–496.
[11] G. Kaplan and D. Levy, Multiplicities in Sylow Sequences and the Solvable Radical, Bull. Austral. Math. Soc., 78 (2008) 477–486.
[12] G. Kaplan and D. Levy, Sylow Products and the Solvable Residual, Comm. Algebra, 36 (2008) 851–857.
[13] G. Kaplan and D. Levy, Products of Sylow Subgroups and the Solvable Radical in Groups St., Andrews 2005, 2, Cambridge University Press, 2007 527–530.
[14] D. Levy, The Average Sylow Multiplicity Character and Solvability of Finite Groups, Comm. Algebra, 38 (2010) 632–644.
[15] D. Levy, The Kernel of the Average Sylow Multiplicity Character and the Solvable Radical, Comm. Algebra, 38 (2010) 4144–4154.
[16] D. Levy, The Average Sylow Multiplicity Character and the Solvable Residual, Comm. Algebra, 41 (2013) 3090–3097.
[17] G. A. Miller. The product of two or more groups, Bull. Amer. Math. Soc., 19 (1913) 303–310.
[18] J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc., 74 (1968) 383–437.
International Journal of Group Theory
Volume 7, Issue 2 - Serial Number 2
Proceedings of the Ischia Group Theory 2016-Part II
June 2018
Pages 1-8

Files

History

  • Receive Date: 23 November 2016
  • Revise Date: 15 January 2017
  • Accept Date: 16 January 2017
  • Published Online: 01 June 2018

Share

How to cite

Statistics