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 Γ(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 Γ(G). For a finite group
G, we show that Γ(G) is a connected graph with diameter at most
two, and     Γ(G) is a complete graph if
and only if G is a p-group for some prime number p. Furthermore,
it is shown that Γ(G)=K5 if and only if either
GCp5,C3×C3, C2×C4 or GQ8.

Keywords


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