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 Γ ⁢ G of a finite group G is defined as follows: the set of vertices is π ⁢ G , the set of primes dividing the order of G , and two vertices p , q are joined by an edge (we write p ∼ q ) if and only if there exists an element in G of order p ⁢ q . We study the groups G such that the prime graph Γ ⁢ G is a tree, proving that, in this case, π ⁢ G ≤ 8 .

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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  10. LUCIDO, M. S., Prime graph components of finite almost simple groups, Rendiconti del Seminario Matematico dell'Università di Padova, 102 (1999), 1-22. Zbl0941.20008MR1739529
  11. LUCIDO, M. S., The diameter of the prime graph of finite groups, Journal of Group Theory, 2 (1999), 157-172. Zbl0921.20020MR1681526
  12. MAZUROV, V. D., The set of orders elements in a finite group, Algebra and Logic, vol. 33, n. 1 (1994), 49-55. Zbl0823.20024MR1287011
  13. PASSMAN, D. S., Permutation groups, W. A. Benjamin, New York (1968). Zbl0179.04405MR237627
  14. ROBINSON, D. R., A course on the theory of groups, Springer-Verlag, Berlin Heidelberg - New York (1982). Zbl0483.20001
  15. WILLIAMS, J. S., Prime graph components of finite groups, Journal of Algebra, 69 (1981), 487-513. Zbl0471.20013MR617092

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