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Finite groups whose noncyclic graphs have positive genus

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Abstract

For a finite noncyclic group G, let \({\rm Cyc} (G)\) be a set of elements a of G such that \(\langle a, b\rangle\) is cyclic for each b of G. The noncyclic graph of G is a graph with the vertex set \(G\setminus {\rm Cyc} (G)\), having an edge between two distinct vertices x and y if \(\langle x, y\rangle\) is not cyclic. In this paper, we show that, for a fixed nonnegative integer k, there are at most finitely many finite noncyclic groups whose noncyclic graphs have (non)orientable genus k. We also classify the finite noncyclic groups whose noncyclic graphs have (non)orientable genus 1, 2, 3 and 4, respectively.

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Acknowledgement

The authors are grateful to the referee for useful suggestions and comments.

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Authors and Affiliations

  1. School of Science, Xi’an Shiyou University, Xi’an, Shaanxi, 710065, China

    X. Ma

  2. School of Mathematics and Statistics, Nanning Normal University, Nanning, Guangxi, 530299, China

    H. Su

Authors
  1. X. Ma
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  2. H. Su
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Correspondence to H. Su.

Additional information

The first author was supported by the National Natural Science Foundation of China (Grant No. 11801441), and the Young Talent fund of University Association for Science and Technology in Shaanxi, China (Program No. 20190507).

The second author was supported by the National Natural Science Foundation of China (Grant No. 11661013).

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Ma, X., Su, H. Finite groups whose noncyclic graphs have positive genus. Acta Math. Hungar. 162, 618–632 (2020). https://doi.org/10.1007/s10474-020-01033-6

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  • DOI: https://doi.org/10.1007/s10474-020-01033-6

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