A counterexample to Zarrin's conjecture on sizes of finite nonabelian simple groups in relation to involution sizes (Q1729717)

\textit{M. Zarrin} produced in [Arch. Math. 111, No. 4, 349--351 (2018; Zbl 1425.20017)] a counterexample to a conjecture of \textit{M. Herzog} [Proc. Am. Math. Soc. 77, 313--314 (1979; Zbl 0421.20009)] that two finite simple groups having the same number of involutions must have the same order. Zarrin went on and conjectured in turn that two finite simple groups having the same number of involutions and also the same number of elements of an odd prime order \(p\) must have the same order. The author of this short note provides an example showing that even Zarrin's conjecture [loc. cit.] is false.