Abstract
The rank and subdegrees (i.e. the number of orbits of the point stabilizer and their respective lengths) of a transitive permutation group are quite simple and very significant characteristics of the group. In certain cases the knowledge of the rank and subdegrees of a group is sufficient for the resolution of the following problems:
-
(a)
identification of rank 3 graphs [6];
-
(b)
proof of the existence and the nonexistence of distance-transitive graphs [7];
-
(c)
computation of the decomposition of the permutation character of the group into irreducibles by means of the character table.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Andrews, The Theory of Partitions, Addison-Wesley, London, 1976.
R. Ball, Maximal subgroups of the symmetric groups, Trans. Amer. Math. Soc., 121 (1966), 398–407.
I.V. Chuvaeva, On some combinatorial objects which are invariant under the Mathieu group M12, In Methods for Complex Systems Studies, pp. 47–52, Moscow, VNIISI, 1983 [In Russian].
A.M. Cohen, A synopsis of known distance-regular graphs with large diameter, Math. Cent. Afd. Zuevere wisk, 1981, N 168.
I.A. Farad2ev, Constructive Enumeration of Combinatorial Objects, In Algorithmic Investigations in Combinatorics, pp. 3–11, Moscow, Nauka, 1978 [In Russian].
X.L. Hubaut, Strongly regular graphs, Discrete Math. 13 (1975), 357–381.
A.A. Ivanov, Combinatoric-algebraic methods for the investigation of distance-regular graphs, Ph.D. Thesis, Moscow, MFTI, 1984 [In Russian].
A.A. Ivanov, Computation of lengths of orbits of a subgroup in a transitive permutation group, In Methods for Complex Systems Studies, pp. 3–8, Moscow, VNIISI, 1983 [See English translation in this volume].
A.A. Ivanov, M.H. Klin, I.A. Farad2ev, Primitive representations of the nonabelian simple groups of order less than 106, Part I, Preprint, Moscow, VNIISI, 1982 [In Russian].
A.A. Ivanov, M.H. Klin, I.A. Farad2ev, Primitive representations of the nonabelian simple groups of order less than 106, Part II, Preprint, Moscow, VNIISI, 1984 [In Russian].
A.A. Ivanov, M.Kh. Klin, S.V. Tsaranov, S.V. Shpektorov, On the problem of computing the subdegrees of transitive permutation groups, Russian Math. Surveys, 38 (1983), 123–124.
A.V. Ivanov, I.A. Farad2ev, Constructive enumeration of incidence systems, Rostock Math. Kolloq., 24 (1983), 4–22 [In Russian].
M.H. Klin, Investigations of algebras of invariant relations for certain classes of permutation groups, Ph. D. Thesis, Nikolaev, 1974 [In Russian].
J. Saxl, On multiplicity free permutation representations, London Math. Soc. Lect, Note Ser., 49 (1981), 337–353.
V.A. Ustimenko-Bakumovskii, Algorithms for the construction of block-designs and symmetric graphs with given automorphism group, In Computation in Algebra and Combinatorics, pp. 137–148, Kiev, 1978 [In Russian].
V.A. Vishenskii, Action of symmetric groups on partitions, Preprint 84.3. Mathematical Institute, Kiev, 1984, pp. 45–53 [In Russian].
V.A. Zaichenko, I.A. Farad2ev, An algorithm for testing the canonicity of incidence systems, In Algorithmic Investigations in Combinatorics, pp. 126–129, Moscow, Nauka, 1978 [In Russian].
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Faradžev, I.A., Ivanov, A.V. (1994). Ranks and Subdegrees of the Symmetric Groups Acting on Partitions. In: Faradžev, I.A., Ivanov, A.A., Klin, M.H., Woldar, A.J. (eds) Investigations in Algebraic Theory of Combinatorial Objects. Mathematics and Its Applications, vol 84. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1972-8_8
Download citation
DOI: https://doi.org/10.1007/978-94-017-1972-8_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4195-1
Online ISBN: 978-94-017-1972-8
eBook Packages: Springer Book Archive