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SOLVABILITY OF FINITE GROUPS VIA CONDITIONS ON PRODUCTS OF 2-ELEMENTS AND ODD p-ELEMENTS

Part of: Representation theory of groups

Published online by Cambridge University Press:  26 April 2010

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: [email protected])
DAN LEVY
Affiliation:
The School of Computer Sciences, The Academic College of Tel-Aviv-Yaffo, 2 Rabenu Yeruham St., Tel-Aviv 61083, Israel
*
✉For correspondence; e-mail: [email protected]
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Abstract

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We observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc.12 (1911)] and Hall [A characteristic property of soluble groups, J. London Math. Soc.12 (1937)] and proved by Thompson [Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc.74(3) (1968)], can be sharpened as follows: a finite group is nonsolvable if and only if it has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p′-elements, a formation Pp,p′, for each prime p. We show that P2,2′ (which contains the odd-order groups) is properly contained in the solvable formation.

Keywords

solvable groupsconjugate elementsformations

MSC classification

Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 2 , October 2010 , pp. 265 - 273
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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