Abstract
In this paper, we consider the possible types of regular maps of order \(2^n\), where the order of a regular map is the order of automorphism group of the map. For \(n \le 11\), M. Conder classified all regular maps of order \(2^n\). It is easy to classify regular maps of order \(2^n\) whose valency or covalency is 2 or \(2^{n-1}\). So we assume that \(n \ge 12\) and \(2\le s,t\le n-2\) with \(s\le t\) to consider regular maps of order \(2^n\) with type \(\{2^s, 2^t\}\). We show that for \(s+t\le n\) or for \(s+t>n\) with \(s=t\), there exists a regular map of order \(2^n\) with type \(\{2^s, 2^t\}\), and furthermore, we classify regular maps of order \(2^n\) with types \(\{2^{n-2},2^{n-2}\}\) and \(\{2^{n-3},2^{n-3}\}\). We conjecture that if \(s+t>n\) with \(s<t\), then there is no regular map of order \(2^n\) with type \(\{2^s, 2^t\}\), and we confirm the conjecture for \(t=n-2\) and \(n-3\).
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Acknowledgements
The first two authors were supported by the National Natural Science Foundation of China (11731002, 12011540376, 12011530455, 12161141005) and the 111 Project of China (B16002), and the third author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (2015R1D1A1A09059016) and by the Korea-China bilateral project (2020K2A9A2A060365).
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Hou, DD., Feng, YQ. & Kwon, Y.S. Regular maps of 2-power order. J Algebr Comb 56, 475–492 (2022). https://doi.org/10.1007/s10801-022-01119-0
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DOI: https://doi.org/10.1007/s10801-022-01119-0