Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 4 / 1 / b / Solution

Partition the conjugacy class into its orbits under conjugation by $N$. Since none of its elements centralizes $N$, each orbit has size $[N:C_N(x)]>1$, a power of $p$. All elements of such an orbit have the same image in $G/N$, because conjugation by an element of $N$ does not alter an $N$-coset. Its orbit sum therefore has imagein characteristic $p$. Summing over the orbits proves $[\mathcal{C}]\in \ker \tau$. This does not require $G=N{C}_G(N)$; that additional assumption is used in the block and Cartan conclusions proved in the root Solution.