Some finite groups with divisibility graph containing no triangles

Document Type : Research Paper

Authors

1 School of Mathematics, Iran University of science and Technology, Tehran, Iran

2 School of Mathematics, Iran University of Science and Technology, Tehran, Iran.

10.29252/as.2019.1483

Abstract

Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $\sigma^{\ast}(G)=2$, a finite simple group $G$ that satisfies the one-prime power hypothesis, a group of type($A$),($B$) or ($C$) and certain metacyclic $p-$groups and a minimal non-metacyclic $p-$group where $p$ is a prime number. We will show that the divisibility graph $D(G)$ for all of them has no triangles.

Keywords


[1] A. Abdolghafourian and M. A. Iranmanesh, On the divisibility graph for nite sets of positive integers, Rocky Mountain. J. Math. Vol. 46 No. 6 (2016), pp. 1755-1770.
[2] A. Abdolghafourian and M. A. Iranmanesh, On the number of connected components of divisibility graph for certain simple groups, Trans. Comb. Vol. 5 No. 2 (2016), pp. 33-40.
[3] A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra Vol. 298 No. 2 (2006), pp. 468-492.
[4] S. Akbari, H. R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra Vol. 270 No. 1 (2003), pp. 169-180.
[5] A. Beltran, M. J. Felipe and C. Melchor, Triangles in the graph of conjugacy classes of normal subgroups, Monatsh Math. Vol. 82 No. 1 (2016), pp. 5-21.
[6] E. A. Bertram, Some applications of graph theory to nite groups, Discrete Math. Vol. 44 No. 1 (1983), pp. 31-43.
[7] E. A. Bertram, M. Herzog and A. Mann, On a graph related to conjugacy classes of groups, Bull. London Math. Soc. Vol. 22 No. 6 (1990), pp. 569-575.
[8] N. Blackburn, Generalizations of certain elementary theorems on p-groups, Proc. London Math. Soc. Vol.11 (1961), pp. 1-22.
[9] D. Bubboloni, S. Dol , M. A. Iranmanesh and C. E. Praeger, On bipartite divisor graphs for group conjugacy class sizes, J. Pure Appl. Algebra Vol. 213 (2009), pp. 1722-1734.
[10] A. R. Camina and R. D. Camina, The influence of conjugacy class sizes on the structure of nite groups: a survey, Asian-Eur. J. Math. Vol.4 No. 4 (2011), pp. 559-588.
[11] C. Casolo, Finite groups with small conjugacy classes, Manuscripta Math. Vol. 82 (1994), pp. 171-189.
[12] S. Dol , Arithmetical conditions on the length of the conjugacy classes in nite groups, J. Algebra Vol. 174 (1995), pp. 753-771.
[13] D. Gorenstein, Finite Groups, Chelsea Publishing Company, New York (1980).
[14] L. He'thelyi and B. Kulshammer, Characters, conjugacy classes and centrally large subgroups of p-groups of small rank, J. Algebra Vol. 340 (2011), pp. 199-210.
[15] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin-New York (1967).
[16] M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, Graphs combin. Vol. 26 No. 1 (2010), pp. 65-105.
[17] G. James and M. Liebeck, Representations and Characters of Groups, Cambridge University Press, Cambridge, (1993).
[18] D. Khoshnevis and Z. Mostaghim, Some properties of graph related to conjugacy classes of special linear group SL2(F), Math. Sci. Lett. Vol. 4 No. 2 (2015), pp.153-156.
[19] M. L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math. Vol. 38 No. 1 (2008), pp. 175-211.
[20] B. Taeri, Cycles and bipartite graph on conjugacy class of groups, Rend. Semin. Mat. Univ. Padova Vol. 123 (2010), pp. 233-247.
[21] J. S. Williams, Prime graph components of nite groups, J. Algebra Vol. 69 No. 2 (1981), pp. 487-513.
[22] M. Xu and Q. Zhang, A classi cation of metacyclic 2-groups, Algebra Colloq. Vol.13 No. 1 (2006), pp. 25-34.