A new characterization of simple $K_3$-groups using same-order type

Abstract

Let $G$ be a group, define an equivalence relation $\sim$ as below: $$\forall \ g, h \in G, \ g \sim h \Longleftrightarrow |g| = |h|$$ the set of sizes of equivalence classes with respect to this relation is called the same-order type of $G$ and denoted by $\alpha(G)$. And $G$ is said a $\alpha_n$-group if $|\alpha(G)|=n$. Let $\pi(G)$ be the set of prime divisors of the order of $G$. A simple group of $G$ is called a simple $K_n$-group if $|\pi(G)|=n$. We give a new characterization of simple $K_3$-groups using same-order type. Indeed we prove that a nonabelian simple group $G$ has same-order type \{r, m, n, k, l\} if and only if $G \cong PSL(2,q)$, with $q=7, 8$ or $9$. This result generalizes the main results in \cite{KKA}, \cite{Sh} and \cite{TZ1}. Motived by the main result in \cite{TZ1} L. J. Taghvasani and M. Zarrin put the following Conjecture 2.10: \textit{Let $S$ be a nonabelian simple $\alpha_n$-group and $G$ a $\alpha_n$-group such that $|S|=|G|$. Then $S \cong G$}. In this paper with a counterexample we give a negative answer to this question.