Quasirecognition by Prime Graph of the Groups 2D2n(q) Where q &lt; 105

Hossein Moradi,Ali Iranmanesh,Mohammad Darafsheh

doi:10.3390/math6040057

Abstract

Let G be a finite group. The prime graph Γ ( G ) of G is defined as follows: The set of vertices of Γ ( G ) is the set of prime divisors of | G | and two distinct vertices p and p ′ are connected in Γ ( G ) , whenever G contains an element of order p p ′ . A non-abelian simple group P is called recognizable by prime graph if for any finite group G with Γ ( G ) = Γ ( P ) , G has a composition factor isomorphic to P. It is been proved that finite simple groups 2 D n ( q ) , where n ≠ 4 k , are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2 D 2 k ( q ) , where k ≥ 9 and q is a prime power less than 10 5 .

Highlights

The prime graph of the Gruenberg-Kegel graph of G is denoted by Γ( G ) and it is a graph with vertex set π ( G ) in which two distinct vertices p and q are adjacent if and only if pq ∈ π ( G ), and in this case we will write p ∼ q
A finite nonabelian simple group P is called quasirecognizable by prime graph if every finite group G
A finite simple nonabelian group P is considered to be quasirecognizable by the set of element orders if each finite group H with πe ( H ) = πe ( P) has a composition factor isomorphic to P

Summary

Introduction

A finite nonabelian simple group P is called quasirecognizable by prime graph if every finite group G A finite simple nonabelian group P is considered to be quasirecognizable by the set of element orders if each finite group H with πe ( H ) = πe ( P) has a composition factor isomorphic to P. Quasirecognizability by prime graph of groups G2 (32n+1 ) and 2 B2 (22n+1 ) has been proved in [4].

Results

Conclusion