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An application of p-factorization methods to symmetric graphs

Published online by Cambridge University Press:  24 October 2008

Richard Weiss
Richard Weiss
Affiliation:
Freie Universität Berlin, W. Germany
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Let Γ be an undirected graph and G a subgroup of aut (Γ) acting transitively on the vertex set V(Γ) of Γ. Let x be an arbitrary vertex of Γ. We denote by T(x) the set of vertices adjacent to x and by G(x)Γ(x) the permutation group induced by the stabilizer G(x) of x in G on Γ(x); G(x)Γ(x) is called the subconstituent of G (with respect to Γ). Let G1(x) = {a ∈ G(x)|a ∈ G(y) for each y ∈ Γ(x)}. For each y ∈ Γ(x), let G(x, y) = G(x) ∩ G(y) and G1(x, y) = G1(x) ∩ G1(y). An s-path is an (s+ l)-tuple (x0, x1, …, xs) of vertices such that xi−1 ∈ Γ(xi) if 1 ≤ i ≤ s and xi−2 ≠ xi if 2 ≤ i ≤ s. Γ is called (G, s)-transitive if G acts transitively on the set of all s-paths but intransitively on the set of all (S+1)-paths in Γ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 1 , January 1979 , pp. 43 - 48
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(1)Bürker, M. and Knapp, W.Zur Vermutung von Sims über primitive Permutations-gruppen II. Arch. Math. 27 (1976), 352–359.CrossRefGoogle Scholar
(2)Dempwolff, U.A factorization lemma and an application. I and II. Arch. Math. 27 (1976), 18–21, 476–479.CrossRefGoogle Scholar
(3)Gardiner, A.Arc transitivity in graphs. Quart. J. Math. Oxford Ser. (2) 24 (1973), 399–407.CrossRefGoogle Scholar
(4)Glauberman, G.Weakly closed elements of Sylow subgroups. Math. Z. 107 (1968), 1–20.CrossRefGoogle Scholar
(5)Glauberman, G.A characteristic subgroup of a p-stable group. Canad. J. Math. 20 (1968), 1101–1135.CrossRefGoogle Scholar
(6)Glauberman, G.Failure of factorization in p-solvable groups. Quart. J. Math. Oxford Ser. (2) 24 (1973), 71–77.CrossRefGoogle Scholar
(7)Gorenstein, D.Finite Groups (New York, Harper & Row, 1968).Google Scholar
(8)Huppert, B.Endliche Gruppen, vol. I (Berlin-Heidelberg-New York, Springer-Verlag, 1967).CrossRefGoogle Scholar
(9)McLaughlin, J.Some groups generated by transvections. Arch. Math. 18 (1967), 363–368.CrossRefGoogle Scholar
(10)McLaughlin, J.Some subgroups of SLn(F 2). Illinois J. Math. 13 (1969), 108–115.Google Scholar
(11)Quirin, W. L.Primitive permutation groups with small orbitals. Math. Z. 122 (1971), 267–274.CrossRefGoogle Scholar
(12)Thompson, J. G.Factorization of p-solvable groups. Pacific J. Math. 16 (1966), 371–372.CrossRefGoogle Scholar
(13)Tutte, W. T.A family of cubical graphs. Proc. Cambridge Philos. Soc. 43 (1947), 459–474.CrossRefGoogle Scholar
(14)Weiss, R.Über symmetrische Graphen und die projektiven Gruppen. Arch. Math. 28 (1977), 110–112.CrossRefGoogle Scholar
(15)Weiss, R.Symmetrische Graphen mit auflösbaren Stabilisatoren. J. Algebra 53 (1978), 412–415.CrossRefGoogle Scholar
(16)Zsigmondy, K.Zur Theorie der Potenzreste. Monatsh. Math. u. Phys. 3 (1892), 265–284.CrossRefGoogle Scholar