On the diameter of the intersection graph of a finite simple group

Xuanlong Ma

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 2, page 365-370
  • ISSN: 0011-4642

Abstract

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Let G be a finite group. The intersection graph Δ G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G , and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1 , where 1 denotes the trivial subgroup of order 1 . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28 . In particular, the intersection graph of a finite non-abelian simple group is connected.

How to cite

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Ma, Xuanlong. "On the diameter of the intersection graph of a finite simple group." Czechoslovak Mathematical Journal 66.2 (2016): 365-370. <http://eudml.org/doc/280105>.

@article{Ma2016,
abstract = {Let $G$ be a finite group. The intersection graph $\Delta _G$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$, and two distinct vertices $X$ and $Y$ are adjacent if $X\cap Y\ne 1$, where $1$ denotes the trivial subgroup of order $1$. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound $28$. In particular, the intersection graph of a finite non-abelian simple group is connected.},
author = {Ma, Xuanlong},
journal = {Czechoslovak Mathematical Journal},
keywords = {intersection graph; finite simple group; diameter},
language = {eng},
number = {2},
pages = {365-370},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the diameter of the intersection graph of a finite simple group},
url = {http://eudml.org/doc/280105},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Ma, Xuanlong
TI - On the diameter of the intersection graph of a finite simple group
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 365
EP - 370
AB - Let $G$ be a finite group. The intersection graph $\Delta _G$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$, and two distinct vertices $X$ and $Y$ are adjacent if $X\cap Y\ne 1$, where $1$ denotes the trivial subgroup of order $1$. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound $28$. In particular, the intersection graph of a finite non-abelian simple group is connected.
LA - eng
KW - intersection graph; finite simple group; diameter
UR - http://eudml.org/doc/280105
ER -

References

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  9. Sawabe, M., 10.3836/tjm/1202136676, Tokyo J. Math. 30 (2007), 293-304. (2007) Zbl1151.20008MR2376509DOI10.3836/tjm/1202136676
  10. Shen, R., 10.1007/s10587-010-0085-4, Czech. Math. J. 60 (2010), 945-950. (2010) Zbl1208.20022MR2738958DOI10.1007/s10587-010-0085-4
  11. Williams, J. S., 10.1016/0021-8693(81)90218-0, J. Algebra 69 (1981), 487-513. (1981) Zbl0471.20013MR0617092DOI10.1016/0021-8693(81)90218-0
  12. Zavarnitsine, A. V., 10.1134/S0037446613010060, Sib. Math. J. 54 (2013), 40-46. (2013) Zbl1275.20009MR3089326DOI10.1134/S0037446613010060
  13. Zelinka, B., Intersection graphs of finite abelian groups, Czech. Math. J. 25 (1975), 171-174. (1975) Zbl0311.05119MR0372075

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