Abstract
We characterise connected cubic graphs admitting a vertex-transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.
Highlights
All the graphs and groups considered in this paper are finite
A very useful tool in the theory of group actions on graphs is the normal quotient method (NQM). This is used to study a family of pairs (Γ, G) having certain additional properties, where Γ is a finite graph and G is a subgroup of the automorphism group Aut(Γ) of Γ. (For example, the family consisting of the pairs (Γ, G) where Γ is a finite G-vertex-transitive graph.) The NQM has an impressive pedigree
We describe a variant of the NQM which is sometimes more successful, the abelian normal quotient method (ANQM)
Summary
A very useful tool in the theory of group actions on graphs is the normal quotient method (NQM) This is used to study (and possibly classify) a family of pairs (Γ, G) having certain additional properties, where Γ is a finite graph and G is a subgroup of the automorphism group Aut(Γ) of Γ. Theorem 1.1 is very useful when applying the abelian normal quotient method to 4-valent arc-transitive graphs, as it deals with case 2 as satisfactorily as one could hope for, that is, giving a complete classification of the possible graphs. Much like Theorem 1.1 with respect to 4-valent arc-transitive graphs, Theorem 1.2 will be very useful when applying the abelian normal quotient method to cubic vertextransitive graphs To illustrate this usefulness, we prove the following: Theorem 1.3. Conjecture 1.4 is best possible as it was shown in [3] that f (n) n1/3, for infinitely many values of n
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