Characterization of $\PSL(2,q)$ by the number of singular elements
Rulin Shen, Deyu Yan
TL;DR
The paper introduces a fingerprint for finite groups based on the set of proportions of $r$-singular elements across primes dividing the group order, denoted $μ(G)$. It proves that the simple group ${\operatorname{PSL}}(2,q)$ is uniquely determined by this fingerprint: if $μ(G)=μ({\operatorname{PSL}}(2,q))$ with $q\ge4$, then $G$ is isomorphic to ${\operatorname{PSL}}(2,q)$. The proof combines structural reductions via a largest normal $p'$-subgroup, a reduction to a simple quotient, and an extensive case analysis against the list of finite simple groups (utilizing classifications such as Walter’s results and the ATLAS). This work supports a broader conjecture that the $μ$-profile may identify non-abelian simple groups, highlighting the utility of singular-element statistics in finite group recognition.
Abstract
Given a finite group $G$, let $π(G)$ denote the set of all primes that divide the order of $G$. For a prime $r \in π(G)$, we define $r$-singular elements as those elements of $G$ whose order is divisible by $r$. Denote by $S_r(G)$ the number of $r$-singluar elements of $G$. We denote the proportion $S_r(G)/|G|$ of $r$-singular elements in $G$ by ${μ_r}(G)$. Let $μ(G) := {\{μ_r}(G) | r\in π(G)\}$ be the set of all proportions of $r$-singular elements for each prime $r$ in $π(G)$. In this paper, we prove that if a finite group $G$ has the same set $μ(G)$ as the simple group $\PSL(2,q)$, then $G$ is isomorphic to $\PSL(2,q)$.