A counterexample to Zarrin’s conjecture on sizes of finite nonabelian simple groups in relation to involution sizes

Abstract

Let $$I_n(G)$$In(G) denote the number of elements of order n in a finite group G. In 1979, Herzog (Proc Am Math Soc 77:313–314, 1979) conjectured that two finite simple groups containing the same number of involutions have the same order. In a 2018 paper (Arch Math 111:349–351, 2018), Zarrin disproved Herzog’s conjecture with a counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group such that $$I_2(G)=I_2(S)$$I2(G)=I2(S) and $$I_p(G) =I_p(S)$$Ip(G)=Ip(S) for some odd prime divisor p, then $$|G|=|S|$$|G|=|S|”. In this paper, we give more counterexamples to Herzog’s conjecture. Moreover, we disprove Zarrin’s conjecture.