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A NOTE ON PERMUTATION GROUPS AND THEIR REGULAR SUBGROUPS

Part of: Permutation groups Graph theory

Published online by Cambridge University Press:  01 October 2008

MING-YAO XU
MING-YAO XU*
Affiliation:
Department of Mathematics, Shanxi Normal University, Linfen, Shanxi 041004, People’s Republic of China (email: [email protected]) LMAM, Institute of Mathematics, Peking University, Beijing 100871, People’s Republic of China (email: [email protected])
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Abstract

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In this note we first prove that, for a positive integer n>1 with n≠p or p2 where p is a prime, there exists a transitive group of degree n without regular subgroups. Then we look at 2-closed transitive groups without regular subgroups, and pose two questions and a problem for further study.

Keywords

transitive permutation groupregular subgroupnon-Cayley number

MSC classification

Secondary: 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 2 , October 2008 , pp. 283 - 287
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Babai, László, ‘On the abstract group of automorphisms’, in: Combinatorics, London Mathematical Society Lecture Note Series, 52 (ed. H. N. V. Temperley) (Cambridge University Press, Cambridge, New York, 1981), pp. 1–40.CrossRefGoogle Scholar
[2]Babai, László, ‘Finite digraphs with regular automorphism groups’, Period. Math. Hungar. 11 (1980), 257–270.CrossRefGoogle Scholar
[3]Giudici, Michael, ‘Factorisations of sporadic simple groups’, J. Algebra 304 (2006), 311–323.CrossRefGoogle Scholar
[4]Godsil, Chris D., ‘GRR’s for non-solvable groups’, in: Algebraic Methods in Graph Theory, Vol. I and II (Szeged, 1978), Colloq. Math. Soc. Jannos Bolyai, 25 (North-Holland, Amsterdam, 1981), pp. 221–239.Google Scholar
[5]Marušič, Dragan, ‘Vertex-transitive graphs and digraphs of order p k’, Ann. Discrete Math. 27 (1985), 115–128.Google Scholar
[6]McKay, Brendan D. and Praeger, Cheryl E., ‘Vertex-transitive graphs which are not Cayley graphs, I’, J. Aust. Math. Soc. (A) 56 (1994), 53–63.Google Scholar
[7]McKay, Brendan D. and Praeger, Cheryl E., ‘Vertex-transitive graphs which are not Cayley graphs, II’, J. Graph Theory 22 (1996), 321–334.3.0.CO;2-N>CrossRefGoogle Scholar
[8]Miller, A. A. and Praeger, Cheryl E., ‘Non-Cayley, vertex transitive graphs of order twice the product of two distinct odd primes’, J. Algebraic Combin. 3 (1994), 77–111.CrossRefGoogle Scholar
[9]Iranmanesh, M. A. and Praeger, Cheryl E., ‘On non-Cayley vertex-transitive graphs of order a product of three primes’, J. Combin. Theory Ser. B 81 (2001), 1–19.CrossRefGoogle Scholar
[10]Wielandt, H., Permutation Groups Through Invariant Relations and Invariant Functions (Ohio State University, Columbus, OH, 1969).Google Scholar
[11]Xu, Ming-Yao, ‘Vertex-primitive digraphs of prime-power order are hamiltonian’, Discrete Math. 128 (1994), 415–417.CrossRefGoogle Scholar