A census of small transitive groups and vertex-transitive graphs. (English) Zbl 1528.20004
Summary: We describe two similar but independently-coded computations used to construct a complete catalogue of the transitive groups of degree less than 48, thereby verifying, unifying and extending the catalogues previously available. From this list, we construct all the vertex-transitive graphs of order less than 48. We then present a variety of summary data regarding the transitive groups and vertex-transitive graphs, focusing on properties that seem to occur most frequently in the study of groups acting on graphs. We illustrate how such catalogues can be used, first by finding a complete list of the elusive groups of order at most 47 and then by completely determining which groups of order at most 47 are CI groups.
MSC:
| 20B20 | Multiply transitive finite groups |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 05E18 | Group actions on combinatorial structures |
| 20-08 | Computational methods for problems pertaining to group theory |
Keywords:
transitive permutation group; vertex-transitive graph; computer enumeration; Cayley graph; elusive group; CI-groupOnline Encyclopedia of Integer Sequences:
Number of transitive permutation groups of degree n.Number of vertex-transitive graphs with n nodes.
Number of Cayley graphs on n nodes.
Number of minimal transitive permutation groups of degree n.
References:
| [1] | Bosma, Wieb; Cannon, John; Playoust, Catherine, The Magma algebra system. I. The user language, J. Symb. Comput., 24, 3-4, 235-265 (1997), Computational algebra and number theory (London, 1993) · Zbl 0898.68039 |
| [2] | Cannon, John J.; Holt, Derek F., The transitive permutation groups of degree 32, Exp. Math., 17, 3, 307-314 (2008) · Zbl 1175.20004 |
| [3] | Coutts, Hannah J.; Quick, Martyn; Roney-Dougal, Colva M., The primitive permutation groups of degree less than 4096, Commun. Algebra, 39, 10, 3526-3546 (2011) · Zbl 1234.20001 |
| [4] | Dobson, Edward; Morris, Joy, Quotients of CI-groups are CI-groups, Graphs Comb., 31, 3, 547-550 (2015) · Zbl 1312.05069 |
| [5] | Giudici, Michael, New constructions of groups without semiregular subgroups, Commun. Algebra, 35, 9, 2719-2730 (2007) · Zbl 1155.20001 |
| [6] | Godsil, Chris; Royle, Gordon, Algebraic Graph Theory, Graduate Texts in Mathematics, vol. 207 (2001), Springer-Verlag: Springer-Verlag New York · Zbl 0968.05002 |
| [7] | Hulpke, Alexander, Constructing transitive permutation groups, J. Symb. Comput., 39, 1, 1-30 (2005) · Zbl 1131.20003 |
| [8] | McKay, Brendan D.; Piperno, Adolfo, Practical graph isomorphism, II, J. Symb. Comput., 60, 94-112 (2014) · Zbl 1394.05079 |
| [9] | Miller, G. A., List of transitive substitution groups of degree twelve, Q. J. Math., 28, 193-231 (1896) · JFM 27.0097.03 |
| [10] | Miller, G. A., The Collected Works of George Abram Miller (1959), University of Illinois, 5 volumes |
| [11] | Royle, Gordon F., The transitive groups of degree twelve, J. Symb. Comput., 4, 2, 255-268 (1987) · Zbl 0683.20002 |
| [12] | Royle, Gordon F., An orderly algorithm and some applications in finite geometry, Discrete Math., 185, 1-3, 105-115 (1998) · Zbl 0956.68148 |
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.
