THE ORDER GRAPHS OF GROUPS

Payrovi, SH.; Pasebani, H.

THE ORDER GRAPHS OF GROUPS

Document Type : Research Paper

Authors

SH. Payrovi ¹
H. Pasebani ²

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)

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) = K_{5}

if and only if either

G ≅ C_{p^{5}}, C_{3} \times C_{3}

C_{2} \times C_{4}

G ≅ Q_{8}

Keywords

References

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