Khoshnevis, D., Mostaghim, Z. (2019). Some finite groups with divisibility graph containing no triangles. Algebraic Structures and Their Applications, 6(2), 57-65. doi: 10.29252/as.2019.1483

Danial Khoshnevis; Zohreh Mostaghim. "Some finite groups with divisibility graph containing no triangles". Algebraic Structures and Their Applications, 6, 2, 2019, 57-65. doi: 10.29252/as.2019.1483

Khoshnevis, D., Mostaghim, Z. (2019). 'Some finite groups with divisibility graph containing no triangles', Algebraic Structures and Their Applications, 6(2), pp. 57-65. doi: 10.29252/as.2019.1483

Khoshnevis, D., Mostaghim, Z. Some finite groups with divisibility graph containing no triangles. Algebraic Structures and Their Applications, 2019; 6(2): 57-65. doi: 10.29252/as.2019.1483

Some finite groups with divisibility graph containing no triangles

^{1}School of Mathematics, Iran University of science and Technology, Tehran, Iran

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

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.

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