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

Abstract

Let $I_n(G)$ 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 $p_1,\,\cdots ,\,p_k$ of $|G|$ and $|S|$ satisfying $2=p_1<\,\cdots \, and $I_{p_i}(G)=I_{p_i}(S)$ for all $i \in \lbrace 1,\,\cdots ,\,k\rbrace $, we have that $|G|=|S|$?”. This paper resolves Malinowska’s question.