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MULTIPLICITIES IN SYLOW SEQUENCES AND THE SOLVABLE RADICAL

Part of: Representation theory of groups

Published online by Cambridge University Press:  01 December 2008

GIL KAPLAN  and
DAN LEVY
GIL KAPLAN
Affiliation:
The School of Computer Sciences, The Academic College of Tel-Aviv-Yaffo, 2 Rabenu Yeruham St., Tel-Aviv 61083, Israel (email: gilk68@gmail.com)
DAN LEVY*
Affiliation:
The School of Computer Sciences, The Academic College of Tel-Aviv-Yaffo, 2 Rabenu Yeruham St., Tel-Aviv 61083, Israel (email: danlevy@trendline.co.il)
*
✉For correspondence; e-mail: danlevy@trendline.co.il
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Abstract

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A complete Sylow sequence, 𝒫=P1,…,Pm, of a finite group G is a sequence of m Sylow pi-subgroups of G, one for each pi, where p1,…,pm are all of the distinct prime divisors of |G|. A product of the form P1⋯Pm is called a complete Sylow product of G. We prove that the solvable radical of G equals the intersection of all complete Sylow products of G if, for every composition factor S of G, and for every ordering of the prime divisors of |S|, there exist a complete Sylow sequence 𝒫 of S, and g∈S such that g is uniquely factorizable in 𝒫 . This generalizes our results in Kaplan and Levy [‘The solvable radical of Sylow factorizable groups’, Arch. Math.85(6) (2005), 490–496].

Keywords

Sylow productsSylow sequencessolvable radicalSylow multiplicityunique1

MSC classification

Secondary: 20D25: Special subgroups (Frattini, Fitting, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 3 , December 2008 , pp. 477 - 486
Copyright
Copyright © 2009 Australian Mathematical Society

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