Tohoku Mathematical Journal

Richness of Smith equivalent modules for finite gap Oliver groups

Toshio Sumi

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Abstract

Let $G$ be a finite group not of prime power order. Two real $G$-modules $U$ and $V$ are $\mathcal{P}(G)$-connectively Smith equivalent if there exists a homotopy sphere with smooth $G$-action such that the fixed point set by $P$ is connected for all Sylow subgroups $P$ of $G$, it has just two fixed points, and $U$ and $V$ are isomorphic to the tangential representations as real $G$-modules respectively. We study the $\mathcal{P}(G)$-connective Smith set for a finite Oliver group $G$ of the real representation ring consisting of all differences of $\mathcal{P}(G)$-connectively Smith equivalent $G$-modules, and determine this set for certain nonsolvable groups $G$.

Article information

Source
Tohoku Math. J. (2), Volume 68, Number 3 (2016), 457-469.

Dates
Received: 25 February 2014
Revised: 29 January 2015
First available in Project Euclid: 23 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1474652268

Digital Object Identifier
doi:10.2748/tmj/1474652268

Mathematical Reviews number (MathSciNet)
MR3550928

Zentralblatt MATH identifier
1361.57040

Subjects
Primary: 57S17: Finite transformation groups
Secondary: 20C15: Ordinary representations and characters

Keywords
Smith problem tangential representation gap group Oliver groups

Citation

Sumi, Toshio. Richness of Smith equivalent modules for finite gap Oliver groups. Tohoku Math. J. (2) 68 (2016), no. 3, 457--469. doi:10.2748/tmj/1474652268. https://projecteuclid.org/euclid.tmj/1474652268


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