Abstract
Let \(\mathcal {M}={{\mathrm{CM}}}(D_n,X,p)\) be a regular Cayley map on the dihedral group \(D_n\) of order \(2n, n \ge 2,\) and let \(\psi \) be the skew-morphism associated with \(\mathcal {M}\). In this paper it is shown that the kernel \({{\mathrm{Ker}}}\psi \) of the skew-morphism \(\psi \) is a dihedral subgroup of \(D_n\) and if \(n \ne 3,\) then the kernel \({{\mathrm{Ker}}}\psi \) is of order at least 4. Moreover, all \(\mathcal {M}\) are classified for which \({{\mathrm{Ker}}}\psi \) is of order 4. In particular, besides four sporadic maps on 4, 4, 8 and 12 vertices, respectively, two infinite families of non-t-balanced Cayley maps on \(D_n\) are obtained.
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The work was partially supported by the Slovenian–Korean bilateral project, Grant No. BI-KOR/13-14-002. The István Kovács was also supported by the ARRS Grant No. P1-0285. The Young Soo Kwon was supported by the 2014 Yeungnam University Research Grant.
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Kovács, I., Kwon, Y.S. Regular Cayley maps on dihedral groups with the smallest kernel. J Algebr Comb 44, 831–847 (2016). https://doi.org/10.1007/s10801-016-0689-3
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DOI: https://doi.org/10.1007/s10801-016-0689-3