The Orbit Graph of Finite p-Groups and Groups of Order pq

S  M  S  Omer; N  H  Sarmin  and A  Erfanian

doi:10.17485/ijst/2014/v7i12.12

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The Orbit Graph of Finite p-Groups and Groups of Order pq

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Indian Journal of Science and Technology

DOI: 10.17485/ijst/2014/v7i12.12

Year: 2014, Volume: 7, Issue: 12, Pages: 2113–2117

Original Article

The Orbit Graph of Finite p-Groups and Groups of Order pq

S. M. S. Omer^1*, N. H. Sarmin² and A. Erfanian³

1 Department of Mathematics, Faculty of Science, University of Benghazi, Benghazi, Libya; Mohamedsana51@yahoo.com
2 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi, Malaysia, Johor Bahru, Malaysia; nhs@utm.my
3 Department of Mathematics and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

In this paper G denotes a finite p-group and Γ denotes a simple undirected graph. The orbit graph is a graph whose vertices are non-central orbit under group action of G on a set. Two vertices v1 and v2 are adjacent in the graph if v g v 1 2 = where v v 1 2 , , ∈Ω and g G∈ . In this paper, the orbit graph of some finite p-groups and group of order pq, where p and q are relatively prime, is found. The orbit graph is determined for the group in the case that a group acts regularly on itself, acts on itself by conjugation, and acts on a set. Besides, some graph properties are found.

Keywords: Orbit Graph; p-Groups and Group of Order pq