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arXiv · 2306.14046

Computation of the Scharlau Invariant, I

R. Keith Dennis·Paul K. Young
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Abstract

The Scharlau invariant determines whether or not a finite group has a fixed point free representation over a field:\ \ if $0$, yes, otherwise, no. Until now it was known to be one of $0$, $1$, $p$, $p^2$ for $p$ a prime dividing the order of the group. We eliminate $p^2$ as a possibility. Work of Scharlau [Sch] reduces the question to the above list with $p^2$ being possible for the groups $\text{SL}_2({\Bbb Z}_p)$ for $p$ a Fermat prime larger than $5$. A computation using GAP in the Senior Thesis [Y] of the second author solves the problem for $p = 17$. With this motivation, we found a short proof of the result not requiring a computer.

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R. Keith Dennis, Paul K. Young. 2023-07-22. Computation of the Scharlau Invariant, I. https://arxiv.org/abs/2306.14046

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