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ON THE ORDER OF ARC-STABILISERS IN ARC-TRANSITIVE GRAPHS, II

  • GABRIEL VERRET (a1)

Abstract

Let $\Gamma $ be a $G$ -vertex-transitive graph and let $(u,v)$ be an arc of $\Gamma $ . It is known that if the local action $G_v^{\Gamma (v)}$ (the permutation group induced by $G_v$ on $\Gamma (v)$ ) is permutation isomorphic to the dihedral group of degree four, then either $|G_{uv}|$ is ‘small’ with respect to the order of $\Gamma $ or $\Gamma $ is one of a family of well-understood graphs. In this paper, we generalise this result to a wider class of local actions.

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References

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[1]Glauberman, G., ‘Normalizers of $p$-subgroups in finite groups’, Pacific J. Math. 29 (1969), 137144.
[2]Potočnik, P., Spiga, P. & Verret, G., ‘On graph-restrictive permutation groups’, J. Combin. Theory Ser. B 102 (2012), 820831.
[3]Potočnik, P., Spiga, P. & Verret, G., ‘Bounding the order of the vertex-stabiliser in $3$-valent vertex-transitive and $4$-valent arc-transitive graphs’, arXiv:1010.2546v1 [math.CO].
[4]Praeger, C. E. & Xu, M. Y., ‘A characterization of a class of symmetric graphs of twice prime valency’, European J. Combin. 10 (1989), 91102.
[5]Tutte, W. T., ‘A family of cubical graphs’, Proc. Cambridge Philos. Soc. 43 (1947), 459474.
[6]Tutte, W. T., ‘On the symmetry of cubic graphs’, Canad. J. Math. 11 (1959), 621624.
[7]Verret, G., ‘On the order of arc-stabilizers in arc-transitive graphs’, Bull. Aust. Math. Soc. 80 (2009), 498505.
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