Some finite groups with divisibility graph containing no triangles

Document Type : Research Paper


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

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


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.


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