Several Quantitative Characterizations of Some Specific Groups

Abstract

Let G be a finite group and let π(G) = {p 1 , p 2 , . . . , p k } be the set of prime divisors of |G| for which, is defined as follows: its vertex set is π(G) and two different vertices p i and p j are adjacent by an edge if and only if G contains an element of order p i p j . The degree of a vertexis the vertex set of a connected component of GK(G), then the largest ω-number which divides |G|, is said to be an order component of GK(G). We will say that the problem of OD-characterization is solved for a finite group if we find the number of pairwise non-isomorphic finite groups with the same order and degree pattern as the group under study. The purpose of this article is twofold. First, we completely solve the problem of OD-characterization for every finite non-abelian simple group with orders having prime divisors at most 29. In particular, we show that there are exactly two non-isomorphic finite groups with the same order and degree pattern as U 4 (2). Second, we prove that there are exactly two non-isomorphic finite groups with the same order components as U 5 (2).