Herzog's conjecture on the number of involutions

M. Herzog proposed the conjecture in 1979: equal numbers of involutions should force equal group orders. The assertion is now contradicted by a claimed explicit pair of finite simple groups.

May 29, 2018 counterexample

M. Zarrin's paper A Counterexample to Herzog's Conjecture on the Number of Involutions claims

$$I_2(\operatorname{PSp}(4,3))=I_2(\operatorname{PSL}(3,4))=315,$$

while

$$|\operatorname{PSp}(4,3)|=25920\ne 20160=|\operatorname{PSL}(3,4)|.$$

Thus the conjecture is false if the computation is accepted. The paper also records the earlier motivating example involving $A_8$ and $\operatorname{PSL}(3,4)$, where both the group orders and involution counts agree.

Current status (as of September 2026): Zarrin's published preprint claims a counterexample, so the conjecture is regarded here as refuted but remains unverified by this scan; no later challenge, withdrawal, or independent verification was found.