A sectional characterization of the Dade group

Details A sectional characterization of the Dade group A sectional characterization of the Dade group

2008-08-28

arxiv.org/abs/0808.3935v1

Journal of Group Theory 11 (2008) 155-183

Mathematics

Group Theory

Scientific paper

10.1515/JGT.2008.010

Let $k$ be a field of characteristic $p$, let $P$ be a finite $p$- group, where $p$ is an odd prime, and let $D(P)$ be the Dade group of endo-permutation $kP$-modules. It is known that $D(P)$ is detected via deflation--restriction by the family of all sections of $P$ which are elementary abelian of rank $\leq2$. In this paper, we improve this result by characterizing $D(P)$ as the limit (with respect to deflation--restriction maps and conjugation maps) of all groups $D(T/S)$ where $T/S$ runs through all sections of $P$ which are either elementary abelian of rank $\leq3$ or extraspecial of order $p^3$ and exponent $p$.

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