On the prime graph of a finite group

doi:10.1017/CBO9780511842474.009

On the prime graph of a finite group

Published online by Cambridge University Press: 05 July 2011

Edited by

G. C. Smith and

Behrooz Khosravi: Affiliation:
Amirkabir University of Technology, Iran; Institute for Research in Fundamental Sciences, Iran
C. M. Campbell: Affiliation:
University of St Andrews, Scotland
M. R. Quick: Affiliation:
University of St Andrews, Scotland
E. F. Robertson: Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal: Affiliation:
University of St Andrews, Scotland
G. C. Smith: Affiliation:
University of Bath
G. Traustason: Affiliation:
University of Bath

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Abstract

Let G be a finite group. The prime graph Г(G) of G is defined as follows. The vertices of Г(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In this paper we give a survey about the question of which groups have the same prime graph. It is proved that some finite groups are uniquely determined by their prime graph. Applications of this result to the problem of recognition of finite groups by the set of element orders are also considered.

Introduction

If n is an integer, then we denote by π(n) the set of all prime divisors of n. If G is a finite group, then the set π(|G|) is denoted by π(G). Also the set of order elements of G is denoted by πe(G). Obviously πe(G) is partially ordered by divisibility. Therefore it is uniquely determined by μ(G), the subset of its maximal elements. We construct the prime graph of G as follows:

Definition 1.1The prime graph Г(G) of a group G is the graph whose vertex set is π(G), and two distinct primes p and q are joined by an edge (we write p ~ q) if and only if G contains an element of order pq. Let t(G) be the number of connected components of Г(G) and let π1(G), π2(G), …, πt(G)(G) be the connected components of Г(G).

Type: Chapter
Information: Groups St Andrews 2009 in Bath , pp. 424 - 428

DOI: https://doi.org/10.1017/CBO9780511842474.009 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2011

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On the prime graph of a finite group

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