Surprising but simple group theory result on conjugacy classes

The number of conjugacy classes is the same as the number of irreducible representations; if $d_i$ is the set of dimensions of the irreducible representations, we know that $d_i | |G|$, hence $d_i$ is odd, and $|G| = \sum d_i^2$. Since $d_i$ is odd, we compute that $d_i^2 \equiv 1, 9 \bmod 16$, so it already follows that $|G| \equiv n \bmod 8$. To get the result $\bmod 16$ it suffices to show that the representations such that $d_i^2 \equiv 9 \bmod 16$ occur an even number of times.

In fact, we claim that no non-trivial irrep is self-dual, from which the above follows (since we can take the dual of an irrep with $d_i^2 \equiv 9 \bmod 16$ to get another one). This is equivalent to the claim that no non-identity element is conjugate to its inverse, which follows from the fact that no element of $G$ can act by a permutation of order two.

This classic theorem of Burnside is often presentated as an example of a theorem that is difficult to prove without using representation theory. However, one can in fact provide elementary proofs for results of this sort. For example, see Reid: The number of conjugacy classes, AMM, 1998, 359-361, where, generalizing results in an earlier 1995 AMM paper by Poonen), if $G$ is a finite group such that every prime divisor $p$ of its order satisfies $p \equiv 1\ (mod\ m)\:,\:$ he proves the strongest possible congruence between $|G|$ and $n$. Below is the Zbl review by R. W. van der Waall (Amsterdam) followed by Burnside's original proof, from S.222, p.294 of Theory of groups of finite order, 1911.

Let $\cal G$ stand in general for a finite group; $p$ for a prime number. Let $m\in\mathbb Z_{\geq 1}$. Consider $${\cal G}_m=\{{\cal G}\ :\ p\mid|{\cal G}|\ \Rightarrow\ p\equiv 1\ (mod\ m)\}.$$ Let $B(m)$ be the greatest common divisor of all numbers $|{\cal G}|-s$, where $\cal G$ runs through ${\cal G}_m$. Here $s$ stands for the number of conjugacy classes of $\cal G$.

In this paper the following is proved. Theorem. If $m >2$, then $B(m)$ is the least common multiple of $48$ and $2m^2$. $\:$ Also $B(2)=16$ and $B(1)=1$.

The proof is elementary, without using representation theory. It uses the facts that each of $3, 16$ and $2m^2$ divides $B(m)$. The case $3\mid B(m)$ is proved here by elementary means; the case $2m^2\mid B(m)$ likewise as done by B. Poonen [in Am. Math. Mon. 102, No. 5, 440-442 (1995; Zbl 0828.11002)]. The fact $16\mid B(m)$ follows for $m >2$ from $B(2)\mid B(m)$, whereas Burnside proved that $16\mid B(2)$ (before 1911).

Reviewer's remark: Using representation theory, Burnside proved that if $|\cal G|$ is odd, then $|\cal G|\equiv s\pmod{16}$. Does there exist an elementary proof, i.e. without using representation theory and without using the Feit-Thompson theorem on the solvability of finite groups of odd order? A result by K. A. Hirsch in that direction seems to be unjustified.