Stabilizers of Half-Arc-Transitive Group Actions on Graphs of Valency 4
by
R. Nedela
Banská Bystrica
Coauthors: D. Marusic (Ljubljana)

We say that a group G acts half-arc-transitively on a graph X if it acts edge- and vertex-transitively but not arc-transitively. Graphs admitting such a group action are necessarily of even valency. The smallest non-trivial case to consider is valency four.

The aim of our talk is to present the following classification of vertex-stabilizers of permutation groups acting half-transitively on 4-valent graphs. It follows that the stabilizers form a particular subclass of class two nilpotent groups.

Theorem Let G be a permutation group G acting half-arc-transitively o na connected 4-valent graph with vertex stabilizer H. Then there are positive integers h and d, 2/3 h <= d <= h and a set of generators <\tau₁, \tau₂, ..., \tau_h> of H satisfying the following relations

(R1) \tau_i²=1, for i=1, ..., h,

(R2) (\tau_i\tau_j)²=1, if 0 < |j-i| < d,

(R3) (\tau_i\tau_j)² = \tau_h-d+i^{\epsilon(j-i, 0)}\tau_h-d+i+1^{\epsilon(j-i, 1)}...\tau_{(h-d+i)+2d-2h+j-1}^{\epsilon(j-i, 2d-2h+j-1)} for all 1 <= i < j <= h,

where j-i >= d and \epsilon(r, s) in {0, 1}.

Furthermore, G=<a, H> for some a in G \H and the conjugation by a sends \tau_i to \tau_i+1 for i=1, 2, ..., h-1.

On the other hand, let (G, H) be a pair of abstract groups satisfying the above conditions. Then the action of G on the set of left cosets of H is faithful with a connected non-self-paired suborbit of length 2 and point stabilizer H, except when H =~ Z₂^h is normal in G, or when h=1, H =~ Z₂ and G is dihedral. Thus the orbital graph X determined by the pair (G, H) is of valency four and G acts on X half-arc-transitively with vertex stabilizer H provided H is not exceptional in the above sense.

Supported by VEGA grant 1/6132/99 and by Ministrstvo za znanost in Technologijo Slovenije.

Date received: May 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafd-18.