Computation of the Scharlau Invariant, I

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 ${\text{SL}}_2({\mathbb{Z}}_p)$ 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.