A new characterization of Mathieu simple groups by the number of singular elements

Given a finite group $G$, let $\pi(G)$ denote the set of all primes that divide the order of $G$. For a prime $p\in \pi(G)$, we define $p$-singular elements as those elements of $G$ whose order is divisible by $p$. We denote the proportion of $p$-singular elements in $G$ by ${\mu_p}(G)$. Let $\mu(G) := {\{\mu_p}(G) | p\in \pi(G)\}$ be the set of all proportions of $p$-singular elements for each prime $p$ that divides $|G|$. In this paper we prove if a finite group $G$ has the same set of proportions as a Mathieu simple group $M$, then $G$ is isomorphic to $M$.