Abstract
Given any subgroup H of a group G, let Γ H (G) be the directed graph with vertex set G such that x is the initial vertex and y is the terminal vertex of an edge if and only if x ≠ y and \({xy\in H}\) . Furthermore, if \({xy\in H}\) and \({yx\in H}\) for some \({x,y\in G}\) with x ≠ y, then x and y will be regarded as being connected by a single undirected edge. In this paper, the structure of the connected components of Γ H (G) is investigated. All possible components are provided in the cases when |H| is either two or three, and the graph Γ H (G) is completely classified in the case when H is a normal subgroup of G and G/H is a finite abelian group.
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Acknowledgments
This research was begun while J. Fasteen participated in an NSF sponsored REU program at the University of Tennessee during the summer of 2002. We would like to thank the referees for carefully reviewing this article. Their suggestions improved the quality of this paper.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Anderson, D.F., Fasteen, J. & LaGrange, J.D. The subgroup graph of a group. Arab. J. Math. 1, 17–27 (2012). https://doi.org/10.1007/s40065-012-0018-1
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DOI: https://doi.org/10.1007/s40065-012-0018-1