An infinitude of counterexamples to Herzog’s conjecture on involutions in simple groups

Abstract In 1979, Herzog conjectured that two finite simple groups containing the same number of involutions have the same order. Zarrin, in a 2018 published paper, disproved Herzog’s conjecture with a counterexample. The goal of this article is to prove that there are infinitely many counterexamples to Herzog’s conjecture. In doing so, we obtain an explicit formula for the number of involutions in the groups involved.


Introduction
We write I n ðGÞ for the number of elements of order n in a finite group G. Herzog [4] conjectured that two finite simple groups containing the same number of involutions have the same order. Zarrin [6] disproved Herzog's conjecture with the following counterexample: I 2 ðPSpð4, 3ÞÞ ¼ 315 ¼ I 2 ðPSLð3, 4ÞÞ, but jPSpð4, 3Þj ¼ 25920 > 20160 ¼ jPSLð3, 4Þj: Then, in view of Herzog's conjecture and Conjecture 2.10 of [5], he conjectured that "if S is a non-abelian simple group and G a group such that I 2 ðGÞ ¼ I 2 ðSÞ and I p ðGÞ ¼ I p ðSÞ for some odd prime divisor p, then jGj ¼ jSj:" Zarrin's conjecture was disproved by the first author in [1], with the following counterexample: , 9ÞÞ and I 13 ðPSLð4, 3ÞÞ ¼ 1866240 ¼ I 13 ðPSLð3, 9ÞÞ; but jPSLð4, 3Þj ¼ 6065280 < 42456960 ¼ jPSLð3, 9Þj: The goal of this short note is to show that there are infinitely many counterexamples to Herzog's conjecture. Moreover, we give fourteen (14) new counterexamples to Zarrin's conjecture, and conclude with a conjecture that there are infinitely many counterexamples to Zarrin's conjecture.

Main results
We begin by outlining the main results of this study. Theorem 2.1 is that there are infinitely many counterexamples to Herzog's conjecture. In proving the result, we obtain explicit formulae (see Equations 18 and 19) for the number of involutions in the groups involved. We give fourteen (14) new counterexamples to Zarrin's conjecture in Table 1, and conclude with a conjecture (see Conjecture 2.5) on the infinitude of counterexamples to Zarrin's conjecture.
Theorem 2.1. There are infinitely many counterexamples to Herzog's conjecture.
Finally, Herzog's conjecture was motivated by the fact that A 8 and PSL (3,4) have the same number of involutions. But jA 8 j ¼ 20160 ¼ jPSLð3, 4Þj; although A 8 6 ffi PSLð3, 4Þ: That was the reason why he conjectured that two finite nonabelian simple groups of different sizes cannot have the same number of involutions. It will be surprising for him and many others to see that PSLðm, 4Þ and higher orders of a certain group "Xðþ1, 2m, 2Þ" (which is isomorphic to A 8 for m ¼ 3) can be used (see Equation (21)) to get infinitely many counterexamples to his conjecture.
Next, we refer to Table 1 for new counterexamples to Zarrin's conjecture. We now give some observations on equalities that cannot hold in general on the number of involutions that coincide with groups that were used in giving more counterexamples to Zarrin's conjecture above; so as to clear any dilemma of thought, on these, for the reader. (ii) For all odd prime powers q > 3, I 2 ðPSUð4, qÞÞ is not always equal to I 2 ðPSLð3, q 2 ÞÞ: An example is Motivated by these insights, we give the following conjecture: Conjecture 2.5. There are infinitely many counterexamples to Zarrin's conjecture. In particular, at least one of the following is true: i. given an integer m ! 4, there exists an odd prime p 1 such that I p 1 ðXðÀ1, 2m, 2ÞÞ ¼ I p 1 ðPSLðm, 4ÞÞ 6 ¼ 0; ii. given an odd integer m > 4, I p 2 ðXðþ1, 2m, 2ÞÞ ¼ I p 2 ðPSLðm, 4ÞÞ, where p 2 is the highest prime divisor of jPSLðm, 4Þj: The counterexamples 11, 12, 13 and 14 given for Zarrin's conjecture in Table 1 above verify Conjecture 2.5(i) for m ¼ 4, 5, 6 and 7 respectively. In the same vein, numbers 9 and 10 of the same Table 1 verify Conjecture 2.5(ii) for m ¼ 5 and m ¼ 7. Finally, Conjecture 2.5(i) together with Equation (20) are aimed at giving infinitely many counterexamples to Zarrin's conjecture; the same is targeted by Conjecture 2.5(ii) and Equation (21).

Funding
The first author is supported by both TU Graz (R-1501000001) and partial funding from the Austrian Science