THE ORDER GRAPHS OF GROUPS

Document Type : Research Paper

Authors

1 Imam Khomeini International University, Qazvin - IRAN.

2 Imam Khomeini International University, Qazvin, IRAN.

Abstract

Let $G$ be a group. The order graph of $G$ is the (undirected)
graph $\Gamma(G)$,
those whose vertices are non-trivial subgroups of $G$ and two distinct
vertices $H$ and $K$ are adjacent if and only if either
$o(H)|o(K)$ or $o(K)|o(H)$. In this paper, we investigate the
interplay between the group-theoretic properties of $G$ and the
graph-theoretic properties of $\Gamma(G)$. For a finite group
$G$, we show that $\Gamma(G)$ is a connected graph with diameter at most
two, and     $\Gamma(G)$ is a complete graph if
and only if $G$ is a $p$-group for some prime number $p$. Furthermore,
it is shown that $\Gamma(G)=K_5$ if and only if either
$G\cong C_{p^5}, C_3\times C_3$, $C_2\times C_4$ or $G\cong Q_8$.

Keywords

References

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