Orientably-regular \(p\)-maps and regular \(p\)-maps. (English) Zbl 1512.05108
Summary: A map is called a \(p\)-map if it has a prime \(p\)-power number of vertices. An orientably-regular (resp. A regular) \(p\)-map is called solvable if the group \(G^+\) of all orientation-preserving automorphisms (resp. the group \(G\) of automorphisms) is solvable; and called normal if \(G^+\) (resp. \(G)\) contains the normal Sylow \(p\)-subgroup. In this paper, it will be proved that both orientably-regular \(p\)-maps and regular \(p\)-maps are solvable and except for few cases that \(p \in \{2, 3\}\), they are normal. Moreover, nonnormal \(p\)-maps will be characterized and some properties and constructions of normal \(p\)-maps will be given.
MSC:
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 05C60 | Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) |
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
References:
| [1] | Catalano, D. A.; Conder, M. D.E.; Du, S. F.; Kwon, Y. S.; Nedela, R.; Wilson, S. E., Classification of regular embeddings of n-dimensional cubes, J. Algebraic Comb., 33, 2, 215-238 (2011) · Zbl 1222.05032 |
| [2] | Conder, M. D.E.; Du, S. F.; Nedela, R.; Škoviera, M., Regular maps with nilpotent automorphism group, J. Algebraic Comb., 44, 863-874 (2016) · Zbl 1362.57004 |
| [3] | Dixon, J. D.; Mortimer, B., Permutation Groups (1996), Springer-Verlag · Zbl 0951.20001 |
| [4] | Du, S. F.; Kwak, J. H., Nonorientable regular embeddings of graphs of order \(p^2\), Discrete Math., 310, 1743-1751 (2010) · Zbl 1222.05189 |
| [5] | Du, S. F.; Kwak, J. H.; Nedela, R., A classification of regular embeddings of graphs of order a product of two primes, J. Algebraic Comb., 19, 123-141 (2004) · Zbl 1042.05027 |
| [6] | Du, S. F.; Jones, G. A.; Kwak, J. H.; Nedela, R.; Škoviera, M., Regular embeddings of \(K_{n , n}\) where n is a power of 2, I: metacyclic case, Eur. J. Comb., 28, 6, 1595-1609 (2007) · Zbl 1123.05029 |
| [7] | Du, S. F.; Jones, G. A.; Kwak, J. H.; Nedela, R.; Škoviera, M., Regular embeddings of \(K_{n , n}\) where n is a power of 2, II: nonmetacyclic case, Eur. J. Comb., 31, 1946-1956 (2010) · Zbl 1221.05079 |
| [8] | Gardiner, A.; Nedela, R.; Širáň, J.; Škoviera, M., Characterization of graphs which underlie regular maps on closed surfaces, J. Lond. Math. Soc., 59, 1, 100-108 (1999) · Zbl 0940.05037 |
| [9] | Gorenstein, D.; Walter, J. H., The characterization of finite groups with dihedral Sylow 2-subgroups I, J. Algebra, 2, 85-151 (1965) · Zbl 0192.11902 |
| [10] | Guralnick, R. M., Subgroups of prime power index in a simple group, J. Algebra, 81, 304-311 (1983) · Zbl 0515.20011 |
| [11] | Hu, K.; Nedela, R.; Škoviera, M.; Wang, N., Regular embeddings of cycles with multiple edges revisited, Ars Math. Contemp., 8, 177-194 (2015) · Zbl 1318.05018 |
| [12] | Huppert, B., Endliche Gruppen I (1979), Springer: Springer Berlin · Zbl 0412.20002 |
| [13] | James, L. D.; Jones, G. A., Regular orientable imbeddings of complete graphs, J. Comb. Theory, Ser. B, 39, 353-367 (1985) · Zbl 0584.05028 |
| [14] | Jones, G. A., Maps on surfaces and Galois groups, Math. Slovaca, 47, 1-33 (1997) · Zbl 0958.05036 |
| [15] | Jones, G. A., Regular embeddings of complete bipartite graphs: classification and enumeration, Proc. Lond. Math. Soc., 101, 427-453 (2010) · Zbl 1202.05028 |
| [16] | Kwak, J. H.; Kwon, Y. S., Classification of nonorientable regular embeddings of complete bipartite graphs, J. Comb. Theory, Ser. B, 101, 191-205 (2011) · Zbl 1226.05100 |
| [17] | Kwon, Y. S.; Nedela, R., Non-existence of nonorientable regular embeddings of n-dimensional cubes, Discrete Math., 307, 3-5, 511-516 (2007) · Zbl 1109.05034 |
| [18] | Li, C. H.; Širáň, Jozef, Regular maps whose groups do not act faithfully on vertices, edges, or faces, Eur. J. Comb., 26, 521-541 (2005) · Zbl 1056.05078 |
| [19] | Malnič, A.; Nedela, R.; Škovier, M., Regular maps with nilpotent automorphism groups, Eur. J. Comb., 33, 1974-1986 (2012) · Zbl 1252.51013 |
| [20] | Nedela, R.; Škoviera, M., Regular embeddings of canonical double coverings of graphs, J. Comb. Theory, Ser. B, 67, 1-3, 249-277 (1996) · Zbl 0856.05029 |
| [21] | Suzuki, M., Group Theory II (1985), Springer-Verlag: Springer-Verlag New York Berlin Heidelberg Tokyo |
| [22] | Wilson, S. E., Cantankerous maps and rotary embeddings of \(K_n\), J. Comb. Theory, Ser. B, 47, 262-273 (1989) · Zbl 0687.05018 |
| [23] | Zhu, Y. H.; Xu, W. Q.; Du, S. F.; Ma, X. S., On the orientable regular embeddings of order prime-cube, Discrete Math., 339, 1140-1146 (2016) · Zbl 1328.05126 |
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