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# Freyd's Generating Hypothesis for Groups with Periodic Cohomology

Published:2011-05-14
Printed: Mar 2012
• Sunil K. Chebolu,
Department of Mathematics, Illinois State University, Normal, IL 61761, U.S.A.
• J. Daniel Christensen,
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7
• Ján Mináč,
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7
 Format: LaTeX MathJax PDF

## Abstract

Let $G$ be a finite group, and let $k$ be a field whose characteristic $p$ divides the order of $G$. Freyd's generating hypothesis for the stable module category of $G$ is the statement that a map between finite-dimensional $kG$-modules in the thick subcategory generated by $k$ factors through a projective if the induced map on Tate cohomology is trivial. We show that if $G$ has periodic cohomology, then the generating hypothesis holds if and only if the Sylow $p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions that are equivalent to the GH for groups with periodic cohomology.
 Keywords: Tate cohomology, generating hypothesis, stable module category, ghost map, principal block, thick subcategory, periodic cohomology
 MSC Classifications: 20C20 - Modular representations and characters 20J06 - Cohomology of groups 55P42 - Stable homotopy theory, spectra

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