Groups in which the prime graph is a tree

Maria Silvia Lucido

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 1, page 131-148
  • ISSN: 0392-4041

Abstract

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The prime graph of a finite group is defined as follows: the set of vertices is , the set of primes dividing the order of , and two vertices , are joined by an edge (we write ) if and only if there exists an element in of order . We study the groups such that the prime graph is a tree, proving that, in this case, .

How to cite

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Lucido, Maria Silvia. "Groups in which the prime graph is a tree." Bollettino dell'Unione Matematica Italiana 5-B.1 (2002): 131-148. <http://eudml.org/doc/194591>.

@article{Lucido2002,
abstract = {The prime graph $\Gamma(G)$ of a finite group $G$ is defined as follows: the set of vertices is $\pi(G)$, the set of primes dividing the order of $G$, and two vertices $p$, $q$ are joined by an edge (we write $p\sim q$) if and only if there exists an element in $G$ of order $pq$. We study the groups $G$ such that the prime graph $\Gamma(G)$ is a tree, proving that, in this case, $|\pi (G)|\leq 8$.},
author = {Lucido, Maria Silvia},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {prime graphs; numbers of connected components; sets of element orders; finite groups; solvable groups; diameters; Fitting lengths; almost simple groups},
language = {eng},
month = {2},
number = {1},
pages = {131-148},
publisher = {Unione Matematica Italiana},
title = {Groups in which the prime graph is a tree},
url = {http://eudml.org/doc/194591},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Lucido, Maria Silvia
TI - Groups in which the prime graph is a tree
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/2//
PB - Unione Matematica Italiana
VL - 5-B
IS - 1
SP - 131
EP - 148
AB - The prime graph $\Gamma(G)$ of a finite group $G$ is defined as follows: the set of vertices is $\pi(G)$, the set of primes dividing the order of $G$, and two vertices $p$, $q$ are joined by an edge (we write $p\sim q$) if and only if there exists an element in $G$ of order $pq$. We study the groups $G$ such that the prime graph $\Gamma(G)$ is a tree, proving that, in this case, $|\pi (G)|\leq 8$.
LA - eng
KW - prime graphs; numbers of connected components; sets of element orders; finite groups; solvable groups; diameters; Fitting lengths; almost simple groups
UR - http://eudml.org/doc/194591
ER -

References

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