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On the genus of a graph related to the join of subgroups of finite abelian group

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Abstract

Let G be a finite group which is not a cyclic p-group, p is a prime number. The undirected simple graph \(\varGamma (G)\) whose vertices are the proper subgroups of G which are not contained in the Frattini subgroup of G and two vertices \(H_1\) and \(H_2\) are joined by an edge if and only if \(G=\left\langle H_1,H_2 \right\rangle \). In this paper, we classify all finite abelian groups G for which \(\varGamma (G)\) has genus one.

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Acknowledgements

The authors are deeply grateful to the referee for careful reading of the manuscript and helpful suggestions. The work reported here is supported by the INSPIRE programme (IF 140700) of Department of Science and Technology, Government of India for the second author.

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  1. Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, 627 012, India

    K. Selvakumar & M. Subajini

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  1. K. Selvakumar
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  2. M. Subajini
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Correspondence to K. Selvakumar.

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Selvakumar, K., Subajini, M. On the genus of a graph related to the join of subgroups of finite abelian group. Afr. Mat. 30, 1223–1236 (2019). https://doi.org/10.1007/s13370-019-00711-1

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  • DOI: https://doi.org/10.1007/s13370-019-00711-1

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