Computation of the Scharlau Invariant, I

TL;DR

This paper advances the understanding of the Scharlau invariant by eliminating the possibility of it being p^2 for certain groups, providing a computer-free proof for specific cases.

Contribution

It proves that the Scharlau invariant cannot be p^2 for finite groups, simplifying the classification of fixed point free representations, and offers a new proof avoiding computational methods.

Findings

Eliminates p^2 as a possible Scharlau invariant value for certain groups.

Provides a computer-free proof for the case p=17.

Clarifies the conditions under which the invariant takes specific values.

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.