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The Laitinen Conjecture for finite non-solvable groups

Part of: Classical Topics in Algebraic Topology Representation theory of groups

Published online by Cambridge University Press:  05 December 2012

Krzysztof Pawałowski  and
Toshio Sumi
Krzysztof Pawałowski
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland (kpa@amu.edu.pl)
Toshio Sumi
Affiliation:
Faculty of Arts and Science, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan (sumi@artsci.kyushu-u.ac.jp)
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Abstract

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For any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A6) or PΣL(2, 27), the Gnil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).

Keywords

Smith equivalencereal G-moduleOliver groupLaitinen conjecturenon-solvable group

MSC classification

Secondary: 57S17: Finite transformation groups 57S25: Groups acting on specific manifolds 55M35: Finite groups of transformations (including Smith theory) 20C15: Ordinary representations and characters
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 1 , February 2013 , pp. 303 - 336
Copyright
Copyright © Edinburgh Mathematical Society 2012

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