Complete regular dessins and non-metacyclic bicyclic 2-groups with cyclic commutator subgroup
Abstract
We study complete regular dessins with underlying graph 𝒦2e,2f, where e≥ f≥ 2, whose automorphism groups are non-metacyclic. Together with the cases of coprime orders and odd-prime-power orders, this is one of the basic classes needed for Jones's open classification problem for regular dessins of complete bipartite graphs. Within the 2-power case, the present paper treats the subcase in which the commutator subgroup is cyclic; it is the first of three papers devoted to the non-metacyclic case. Our approach is group-theoretic. The automorphism group G of a regular dessin of 𝒦2e,2f admits an exact bicyclic factorization G=⟨ b⟩⟨ w⟩ of type (2e,2f) with ⟨ b⟩∩⟨ w⟩=1. For a fixed group G, two exact generating pairs give isomorphic dessins exactly when they lie in the same orbit under Aut(G). Thus the classification reduces to three group-theoretic tasks: determining the isomorphism classes of the possible groups G, characterizing their exact bicyclic generating pairs, and computing |Aut(G)|. Berkovich and Janko divided the relevant non-metacyclic groups, through parameterized presentations, according to whether the commutator subgroup is cyclic, contains a cyclic subgroup of index $2$, or contains a cyclic subgroup of index 2r with r>1. However, their classification does not identify which parameter values in these presentations define isomorphic groups; this problem was raised by Janko in {J}. We carry out these steps for the cyclic-commutator case. This settles Janko's isomorphism-classification problem in this case and yields a complete classification and enumeration of the corresponding regular dessins.
Related
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BibTeX
@misc{fan2026complete,
title = {{Complete regular dessins and non-metacyclic bicyclic 2-groups with cyclic commutator subgroup}},
author = {Wenwen Fan and Yuan Shi and Yun Tian Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39407},
eprint = {2610.39407},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39407},
}
RIS
TY - UNPB
TI - Complete regular dessins and non-metacyclic bicyclic 2-groups with cyclic commutator subgroup
AU - Wenwen Fan
AU - Yuan Shi
AU - Yun Tian Wu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39407
UR - https://arxiv.org/abs/2610.39407
AB - We study complete regular dessins with underlying graph 𝒦_2^e,2^f, where e≥ f≥ 2, whose automorphism groups are non-metacyclic. Together with the cases of coprime orders and odd-prime-power orders, this is one of the basic classes needed for Jones's open classification problem for regular dessins of complete bipartite graphs. Within the 2-power case, the present paper treats the subcase in which the commutator subgroup is cyclic; it is the first of three papers devoted to the non-metacyclic case. Our approach is group-theoretic. The automorphism group G of a regular dessin of 𝒦_2^e,2^f admits an exact bicyclic factorization G=⟨ b⟩⟨ w⟩ of type (2^e,2^f) with ⟨ b⟩∩⟨ w⟩=1. For a fixed group G, two exact generating pairs give isomorphic dessins exactly when they lie in the same orbit under Aut(G). Thus the classification reduces to three group-theoretic tasks: determining the isomorphism classes of the possible groups G, characterizing their exact bicyclic generating pairs, and computing |Aut(G)|. Berkovich and Janko divided the relevant non-metacyclic groups, through parameterized presentations, according to whether the commutator subgroup is cyclic, contains a cyclic subgroup of index $2$, or contains a cyclic subgroup of index 2^r with r>1. However, their classification does not identify which parameter values in these presentations define isomorphic groups; this problem was raised by Janko in {J}. We carry out these steps for the cyclic-commutator case. This settles Janko's isomorphism-classification problem in this case and yields a complete classification and enumeration of the corresponding regular dessins.
ER -