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

Chimere Stanley Anabanti

doi:10.5802/crmath.130

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 ,⋯,p k of |G| and |S| satisfying 2=p 1 <⋯<p k and I p i (G)=I p i (S) for all i∈{1,⋯,k}, we have that |G|=|S|?”. This paper resolves Malinowska’s question.

Highlights

In 1979, Herzog [6] conjectured that two finite simple groups containing the same number of involutions have the same order
Before we give a proof of Theorem 1, we first recall some important results
The groups (P SL3(4) and P SL4(2) or Ω2n+1(q) and P Sp2n(q) for some odd prime power q, and some n ≥ 3) mentioned in Theorem 2 above were given in Artin’s paper since they were known at that time

Summary

Introduction

In 1979, Herzog [6] conjectured that two finite simple groups containing the same number of involutions have the same order. In an attempt to reformulate the mentioned conjecture of Zarrin (considering the works in [1,6,9]), Malinowska [8] 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|?”. We reinstate Malinowska’s question as follows: “What is the smallest positive integer k such that whenever there exist two nonabelian finite simple groups S and G that satisfy Hypothesis(k), we have that |G| = |S|?”.

Proof of Theorem 1

An observation