Skew-morphisms of nonabelian characteristically simple groups. (English) Zbl 1542.20048
Summary: A skew-morphism of a finite group \(G\) is a permutation \(\sigma\) on \(G\) fixing the identity element such that the product of \(\langle\sigma\rangle\) with the left regular representation of \(G\) forms a permutation group on \(G\). This permutation group is called the skew-product group of \(\sigma\). The skew-morphism was introduced as an algebraic tool to investigate regular Cayley maps. In this paper, we characterize skew-products of skew-morphisms of finite nonabelian characteristically simple groups (see Theorem 1.2) and the corresponding regular Cayley maps (see Theorem 1.6).
MSC:
| 20C15 | Ordinary representations and characters |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
Software:
MagmaReferences:
| [1] | Bachratý, M.; Jajcay, R., Powers of skew-morphisms, (Symmetries in Graphs, Maps, and Polytopes. Symmetries in Graphs, Maps, and Polytopes, Springer Proceedings in Mathematics & Statistics, vol. 159 (2016)), 1-26 · Zbl 1354.05064 |
| [2] | Bachratý, M.; Jajcay, R., Classification of coset-preserving skew-morphisms of finite cyclic groups, Australas. J. Comb., 67, 259-280 (2017) · Zbl 1375.05126 |
| [3] | Bachratý, M.; Conder, M.; Verret, G., Skew product groups for monolithic groups (2019) |
| [4] | Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system. I. The user language, J. Symb. Comput., 24, 3-4, 235-265 (1997) · Zbl 0898.68039 |
| [5] | Conder, M.; Jajcay, R.; Tucker, T., Cyclic complements and skew-morphisms of groups, J. Algebra, 453, 68-100 (2016) · Zbl 1338.20019 |
| [6] | Conder, M.; Tucker, T., Regular Cayley maps for cyclic groups, Trans. Am. Math. Soc., 366, 3585-3609 (2014) · Zbl 1290.05160 |
| [7] | Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of Finite Groups (1985), Clarendon Press: Clarendon Press Oxford · Zbl 0568.20001 |
| [8] | d’Azevedo, A. B.; Catalano, D. A.; Širáň, J., Bi-rotary maps of negative prime characteristic, Ann. Comb., 23, 1, 27-50 (2019) · Zbl 1414.05090 |
| [9] | Du, S. F.; Hu, K., Skew-morphisms of cyclic 2-groups, J. Group Theory, 22, 4, 617-635 (2019) · Zbl 1466.20016 |
| [10] | Du, S. F.; Wang, F. R., Arc-transitive cubic Cayley graphs on \(\operatorname{PSL}(2, p)\), Sci. China Ser. A, 48, 1297-1308 (2005) · Zbl 1124.05041 |
| [11] | Jajcay, R.; Širáň, J., Skew-morphisms of regular Cayley maps, Discrete Math., 224, 167-179 (2002) · Zbl 0988.05047 |
| [12] | Jones, G.; Singerman, D., Theory of maps on orientable surfaces, Proc. Lond. Math. Soc. (3), 37, 2, 273-307 (1978) · Zbl 0391.05024 |
| [13] | Jones, G., Cyclic regular subgroups of primitive permutation groups, J. Group Theory, 5, 403-407 (2004) · Zbl 1012.20002 |
| [14] | King, C. S.H., Generation of finite simple groups by an involution and an element of prime order, J. Algebra, 478, 153-173 (2017) · Zbl 1376.20018 |
| [15] | Kovács, I.; Nedela, R., Decomposition of skew-morphisms of cyclic groups, Ars Math. Contemp., 4, 329-349 (2011) · Zbl 1238.05283 |
| [16] | Kovács, I.; Marušič, D.; Muzychuk, M. E., On G-arc-regular dihedrants and regular dihedral maps, J. Algebraic Comb., 38, 437-455 (2013) · Zbl 1271.05047 |
| [17] | Kovács, I.; Kwon, Y. S., Regular Cayley maps on dihedral groups with smallest kernel, J. Algebraic Comb., 44, 831-847 (2016) · Zbl 1352.05194 |
| [18] | Kovács, I.; Kwon, Y. S., Regular Cayley maps for dihedral groups, J. Comb. Theory, Ser. B, 148, 84-124 (2021) · Zbl 1459.05119 |
| [19] | Kovács, I.; Nedela, R., Skew-morphisms of cyclic p-groups, J. Group Theory, 20, 6, 1135-1154 (2017) · Zbl 1388.20037 |
| [20] | Kwak, J. H.; Kwon, Y. S., Unoriented Cayley maps, Studia Sci. Math. Hung., 43, 2, 137-157 (2006) · Zbl 1112.05028 |
| [21] | Kwon, Y. S., A classification of regular t-balanced Cayley maps for cyclic groups, Discrete Math., 313, 656-664 (2013) · Zbl 1259.05074 |
| [22] | Li, C. H.; Praeger, C. E., On finite permutation groups with a transitive cyclic subgroup, J. Algebra, 349, 117-127 (2012) · Zbl 1257.20002 |
| [23] | Malle, G.; Saxl, J.; Weigel, T., Generation of classical groups, Geom. Dedic., 49, 1, 85-116 (1994) · Zbl 0832.20029 |
| [24] | Richter, R. B.; Širáň, J.; Jajcay, R.; Tucker, T. W.; Watkins, M. E., Cayley maps, J. Comb. Theory, Ser. B, 95, 2, 189-245 (2005) · Zbl 1079.05029 |
| [25] | Scott, L. L., Representations in characteristic p, (The Santa Cruz Conference on Finite Groups. The Santa Cruz Conference on Finite Groups, Univ. California, Santa Cruz, Calif., 1979 (1980), Amer. Math. Soc.: Amer. Math. Soc. Providence, R.I.), 319-331 · Zbl 0458.20039 |
| [26] | Suzuki, M., Group Theory I (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0472.20001 |
| [27] | Wielandt, H., Finite Permutation Groups (1964), Academic Press: Academic Press New York-London, x+114 pp. · Zbl 0138.02501 |
| [28] | Zhang, J. Y., A classification of regular Cayley maps with trivial Cayley-core for dihedral groups, Discrete Math., 338, 1216-1225 (2015) · Zbl 1309.05097 |
| [29] | Zhang, J. Y., Regular Cayley maps of skew-type 3 for dihedral groups, Discrete Math., 388, 1163-1172 (2015) · Zbl 1309.05096 |
| [30] | Zhang, J. Y.; Du, S. F., On the skew-morphisms of dihedral groups, J. Group Theory, 19, 993-1016 (2016) · Zbl 1358.20019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
