A finite non-regular primitive permutation group
G
is
extremely primitive
if a point stabiliser acts primitively on each of its nontrivial orbits. Such groups have been studied for almost a century, ...finding various applications. The classification of extremely primitive groups was recently completed by Burness and Lee, who relied on an earlier classification of soluble extremely primitive groups by Mann, Praeger and Seress. Unfortunately, there is an inaccuracy in the latter classification. We correct this mistake, and also investigate regular linear spaces which admit groups of automorphisms that are extremely primitive on points.
Let p be a prime and let L be either the intransitive permutation group Cp×Cp of degree 2p or the transitive permutation group CpwrC2 of degree 2p. Let Γ be a connected G-vertex-transitive and ...G-edge-transitive graph and let v be a vertex of Γ. We show that if the permutation group induced by the vertex-stabiliser Gv on the neighbourhood Γ(v) is isomorphic to L then either |V(Γ)|≥p|Gv|logp(|Gv|/2), or |V(Γ)| is bounded by a constant depending only on p, or Γ is a very-well understood graph. This generalises a few recent results.
Cayley graphs on abelian groups Dobson, Edward; Spiga, Pablo; Verret, Gabriel
Combinatorica (Budapest. 1981),
08/2016, Letnik:
36, Številka:
4
Journal Article
Recenzirano
Odprti dostop
Let
A
be an abelian group and let ι be the automorphism of
A
defined by: ι: a ↦ a
−1
. A Cayley graph Γ = Cay(
A,S
) is said to have an automorphism group
as small as possible
if Aut(Γ)=A⋊<ι>. In ...this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.
In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a ...connected vertex-transitive graph with n vertices and of valence d, d≤4, is at most cdn where c3=1 and c4=9. Whether such a constant cd exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism g of a finite connected 3-valent vertex-transitive graph Γ, Γ≇K3,3, has a regular orbit, that is, an orbit of 〈g〉 of length equal to the order of g. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of g. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms C of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of C.
A Cayley graph for a group G is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of G is an element of the normaliser of G. A group G is then ...said to be CCA if every connected Cayley graph on G is CCA. We show that a finite simple group is CCA if and only if it has no element of order 4. We also show that “many” 2-groups are non-CCA.
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