A counterexample to Herzog's conjecture on the number of involutions (Q667711)

It is shown that the number of involutions of the projective symplectic group of degree 4 over the finite field with 3 elements (denoted by $\mathrm{PSp}(4,3)$) is equal to the number of involutions of the group $\mathrm{PSL}(3,4)$. The common number of involutions is 315. These two groups have orders 25920 and 20160, respectively. Thus they provide a counterexample to a conjecture of \textit{M. Herzog} [Proc. Am. Math. Soc. 77, 313--314 (1979; Zbl 0421.20009)] suggesting that if two finite simple groups have the same number of involutions then they have the same order.