  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.
