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Freyd's Generating Hypothesis for Groups with Periodic Cohomology
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- Journal:
- Canadian Mathematical Bulletin / Volume 55 / Issue 1 / 01 March 2012
- Published online by Cambridge University Press:
- 20 November 2018, pp. 48-59
- Print publication:
- 01 March 2012
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- Article
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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 
$\text{GH}$ for groups with periodic cohomology.
