A question of Malinowska on sizes of finite nonabelian simple groups in relation to involution sizes

Let In (G) denote the number of elements of order n in a finite group G . Malinowska recently asked “what is the smallest positive integer k such that whenever there exist two nonabelian finite simple groups S and G with prime divisors p1, · · · , pk of |G| and |S| satisfying 2 = p1 < ·· · < pk and Ipi (G) = Ipi (S) for all i ∈ {1, · · · , k}, we have that |G| = |S|?”. This paper resolves Malinowska’s question. 2020 Mathematics Subject Classification. 20D60,20D06. Funding. The author is supported by both TU Graz and partial funding from the Austrian Science Fund (FWF): P30934-N35, F05503, F05510. He is also at the University of Nigeria, Nsukka. Manuscript received 25th May 2020, revised 6th October 2020, accepted 7th October 2020.