ChapterZero

I came across this problem from a Putnam exam: Let be a finite group of matrices in such that . Show that .

I haven’t been able to solve it in three days of sporadic efforts, but I’m slowly getting a handle on this: I’ve convinced myself that the are diagonalizable with only in their spectrum.

Solution Sweet! Lieven from Neverendingbooks came up with this neat proof: Note that is a projection onto its eigenspace– this follows from the basic group theoretical fact that . We know that the dimension of the range space of a projection operator is the trace of the operator, so voila!