A new characterization of simple K3-groups using same-order type

Abstract

Let G be a group, define an equivalence relation ∼ as below: ∀ g, h ∈ G, g ∼ h ⇐⇒|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 α(G). And G is said a αn-group if |α(G)| = n. Let π(G) be the set of prime divisors of the order of G. A simple group of G is called a simple Kn-group if |π(G)| = n. We give a new characterization of simple K3-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 ≅ PSL(2,q), with q = 7, 8 or 9. This result generalizes the main results in (4), (6) and (8). Moreover based on the main result in (8) we have the natural question: Let S be a nonabelian simple αn-group and G a αn-group such that |S| = |G|. Then S ≅ G. In this paper with a counterexample we give a negative answer to this question.