= Odd-order group conjugacy-class congruence
{title2=$|G|-k(G)\equiv0\pmod{16}$}
An odd-order <finite group> has only odd irreducible degrees, by <irreducible character degree divides the group order>. Its nontrivial characters occur in distinct conjugate pairs, by the <Frobenius-Schur indicator>. Therefore $|G|-k(G)=\sum_\chi(\chi(1)^2-1)$ is a sum of pairs $2(d^2-1)$ with $d$ odd. Since $d^2\equiv1\pmod8$, each pair is divisible by sixteen. Here $k(G)$ is the number of <conjugacy classes>.
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