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Edge-Transitivity of Cayley Graphs Generated by Transpositions

Ashwin Ganesan (2016)

Discussiones Mathematicae Graph Theory

Let S be a set of transpositions generating the symmetric group Sn (n ≥ 5). The transposition graph of S is defined to be the graph with vertex set {1, . . . , n}, and with vertices i and j being adjacent in T(S) whenever (i, j) ∈ S. In the present note, it is proved that two transposition graphs are isomorphic if and only if the corresponding two Cayley graphs are isomorphic. It is also proved that the transposition graph T(S) is edge-transitive if and only if the Cayley graph Cay(Sn, S) is edge-transitive....

Every reasonably sized matrix group is a subgroup of S ∞

Robert Kallman (2000)

Fundamenta Mathematicae

Every reasonably sized matrix group has an injective homomorphism into the group S of all bijections of the natural numbers. However, not every reasonably sized simple group has an injective homomorphism into S .

Examples of non-shy sets

Randall Dougherty (1994)

Fundamenta Mathematicae

Christensen has defined a generalization of the property of being of Haar measure zero to subsets of (abelian) Polish groups which need not be locally compact; a recent paper of Hunt, Sauer, and Yorke defines the same property for Borel subsets of linear spaces, and gives a number of examples and applications. The latter authors use the term “shyness” for this property, and “prevalence” for the complementary property. In the present paper, we construct a number of examples of non-shy Borel sets...

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.

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