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

2019/11/30 by Lima, Igor, Pereira, Josyane
#20D06 #20D60 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1912.00125

Abstract

Let G be a group, define an equivalence relation ∼ as below: ∀ g, h ∈ G, g ∼ 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 α(G). And G is said a α_n-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 K_n-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 \citeKKA, \citeSh and \citeTZ1. Motived by the main result in \citeTZ1 L. J. Taghvasani and M. Zarrin put the following Conjecture 2.10: Let S be a nonabelian simple α_n-group and G a α_n-group such that |S|=|G|. Then S ≅ G. In this paper with a counterexample we give a negative answer to this question.

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