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Volume 9, Issue 6, December 2020, Page: 175-178
A Construction of Imprimitive Groups of Rank 4 or 5
Chang Wang, School of Mathematics, Yunnan Normal University, Kunming, China
Renbing Xiao, School of Mathematics, Yunnan Normal University, Kunming, China
Received: Sep. 15, 2020;       Accepted: Oct. 23, 2020;       Published: Nov. 4, 2020
Abstract
Let G be a transitive permutation group acting on a finite set Ω. For a point α of Ω, the set of the images of G acting on α is called the orbit of α under G and is denoted by αG, and the set of elements in G which fix α is called the stabilizer of α in G and is denoted by Gα. We can get some new orbits by using the natural action of the stabilizer Gα on Ω, and then we can define the suborbit of G. The suborbits of G on Ω are defined as the orbits of a point stabilizer on Ω. The number of suborbits is called the rank of G and the length of suborbits is called the subdegree of G. For finite primitive groups, the study of the rank and subdegrees of group has a long history. In this paper, we construct a class of imprimitive permutation groups of rank 4 or 5 by using imprimitive action and product action of wreath product, determine the number and the length of the suborbits, and extend the results to imprimitive permutation groups of rank m+1 and 2n+1, where m and n are positive integers.
Keywords
Permutation Group, Transitive Action, Rank, Suborbit
Chang Wang, Renbing Xiao, A Construction of Imprimitive Groups of Rank 4 or 5, Applied and Computational Mathematics. Vol. 9, No. 6, 2020, pp. 175-178. doi: 10.11648/j.acm.20200906.11
Reference
[1]
N. Biggs. Algebraic Graph Theory, second ed. Cambridge Unir. Press, New York, 1992.
[2]
J. D. Dixon and Mortimer, B. Permutation groups. Graduate Texts in Mathematics, Springer-Verlag (1996), Berlin.
[3]
Joanna B. Fawcett, Michael Giudici, Cai Heng Li, Cheryl E. Praeger, Gordon Royle and Gabriel Verret. Primitive permutation groups with a suborbit of length 5 and vertex-primitive graphs of valency 5, Journal of Combinatorial Theory, Series A 157 (2018), 247-266.
[4]
D. G. Higman. Finite permutation groups of rank 3. Math. Z. 86 (1964), 145-156.
[5]
W. Knapp. On the point stabilizer in a primitive permutation group. Math. Z. 133 (1973), 137-168.
[6]
C. H. Li, Z. P. Lu and Dragan Marušič. On primitive permutation groups with small suborbits and their orbital graphs. Journal of Algebra, 279 (2004), 749-770.
[7]
C. H. Li, H. S. Sim. On half-transitive metacirculants of prime-power order. J. Combin. Theory Ser. B 81 (2001) 45-51.
[8]
P. M. Neumann. Finite permutation groups. edge-coloured graphs and matrices, in: Topics in Group Theory and Computation, Proc. Summer school, University College, Galway, (1977), pp. 82-118.
[9]
C. E. Praeger. Primitive permutation groups with a doubly transitive subconstituent. J. Austral. Math. Soc. (scries A) 45 (1988), 66-77.
[10]
W. L. Quirin. Primitive permutation groups with small orbitals. Math. Z. 122 (1971), 267-274.
[11]
C. C. Sims. Graphs and finite permutation groups. Math. Z. 95 (1967), 76-86.
[12]
H. Wielant. Finite Permutation Groups. Academic Press, New York, 1964.
[13]
J. Wang. The primitive permutation groups with an orbital of length 4. Comm. Algebra 20 (1992), 889-921.
[14]
J. Wang. Primitive permutation groups with a solvable subconstituent of degree 5. Beijing Daxue Xuebao Ziran Kexue Ban 31 (1995), 520-526.
[15]
J. Wang. Primitive permutation groups with an unfaithful subconstituent containing A5. Algebra Colloq. 3 (1996), 11-18.
[16]
W. J. Wong. Determination of a class of primitive permutation groups. Math. Z. 99 (1967), 235-246.