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On the prime graph of a finite group

Published online by Cambridge University Press:  05 July 2011

By
Behrooz Khosravi
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
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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Summary

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).

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Type
Chapter
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Groups St Andrews 2009 in Bath , pp. 424 - 428
Publisher: Cambridge University Press
Print publication year: 2011

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References

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