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Extremely primitive groups and linear spaces

Haiyan Guan, Shenglin Zhou (2016)

Czechoslovak Mathematical Journal

A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let 𝒮 be a nontrivial finite regular linear space and G ≤ Aut ( 𝒮 ) . Suppose that G is extremely primitive on points and let rank ( G ) be the rank of G on points. We prove that rank ( G ) ≥ 4 with few exceptions. Moreover, we show that Soc ( G ) is neither a sporadic group nor an alternating group, and G = PSL ( 2 , q ) with q + 1 a Fermat prime if Soc ( G ) is a finite classical simple group.

Finiteness results for Hilbert's irreducibility theorem

Peter Müller (2002)

Annales de l’institut Fourier

Let k be a number field, 𝒪 k its ring of integers, and f ( t , X ) ∈ k ( t ) [ X ] be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations t ↦ t ¯ ∈ 𝒪 k such that f ( t ¯ , X ) is still irreducible. In this paper we study the set Red f ( 𝒪 k ) of those t ¯ ∈ 𝒪 k with f ( t ¯ , X ) reducible. We show that Red f ( 𝒪 k ) is a finite set under rather weak assumptions. In particular, previous results obtained by diophantine approximation techniques, appear as special cases of some of our results. Our method is different. We use elementary group...

Graphs having no quantum symmetry

Teodor Banica, Julien Bichon, Gaëtan Chenevier (2007)

Annales de l’institut Fourier

We consider circulant graphs having p vertices, with p prime. To any such graph we associate a certain number k , that we call type of the graph. We prove that for p ≫ k the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.

Hexavalent ( G , s ) -transitive graphs

Song-Tao Guo, Xiao-Hui Hua, Yan-Tao Li (2013)

Czechoslovak Mathematical Journal

Let X be a finite simple undirected graph with a subgroup G of the full automorphism group Aut ( X ) . Then X is said to be ( G , s ) -transitive for a positive integer s , if G is transitive on s -arcs but not on ( s + 1 ) -arcs, and s -transitive if it is ( Aut ( X ) , s ) -transitive. Let G v be a stabilizer of a vertex v ∈ V ( X ) in G . Up to now, the structures of vertex stabilizers G v of cubic, tetravalent or pentavalent ( G , s ) -transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers G v of connected hexavalent ( G , s ) -transitive...

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