Quasiprimitive groups and blow-up decompositions

Baddeley, Robert W.; Praeger, Cheryl E.; Schneider, Csaba

doi:10.1016/j.jalgebra.2006.11.006

Cited by 6 publications

(4 citation statements)

References 15 publications

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“…By Theorem 2.7, it suffices to show, for G ∈ {C 5 , D 5 , AGL(1, 7), PGL(2, 7), PSL (2,8), PΓL(2, 8)}, that there exists some a ∈ T n \ S n such that the semigroup a g | g ∈ G is not generated by idempotents. Using the GAP computational algebra system, it is possible to show that the required transformations a are [1, 3, 2, 2, 2], [1,2,3,3,3], [1,2,3,3,3,3,3], [6,2,3,4,6,6,6,6], [1,2,3,5,4,5,4,4,5], and [1,2,3,5,4,5,4,4,5], respectively; see Section 5 and [2] for further description of the computations.…”

Section: The Proofs Of the Main Resultsmentioning

confidence: 99%

“…Primitive permutation groups are described by the O'Nan-Scott Theorem that divides these group into several classes. Statements of this theorem can be found in [8,Section 4.8] and in [5, Sections 4.4-4.5], while in [4] there is a detailed comparison of the different versions of the theorem that can be found in the literature. Since the order of a non-abelian finite simple group is at least 60, combining the O'Nan-Scott Theorem with the bound in Lemma 4.2 gives that a proper universal transversal group is either an almost simple group, an affine group, or a subgroup of a wreath product in product action.…”

Section: The Classification Of Universal Transversal Groupsmentioning

confidence: 99%

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Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

2011

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Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then G, a \ G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates a g = g −1 ag of a by elements g ∈ G generate a semigroup denoted a g | g ∈ G . We classify the finite permutation groups G on a finite set X such that the semigroups G, a , G, a \G, and a g | g ∈ G are regular for all transformations of X. We also classify the permutation groups G on a finite set X such that the semigroups G, a \ G and a g | g ∈ G are generated by their idempotents for all non-invertible transformations of X.

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Section: The Proofs Of the Main Resultsmentioning

confidence: 99%

Section: The Classification Of Universal Transversal Groupsmentioning

confidence: 99%

Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

2011

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“…The points y such that D G(64,5) (1, y) = 2 are {5, 6,7,8,9,10,11,12,15,17,18,19,20,23,25,29,30,31,32,33,34,35,36,40,42,45,46,47,48,50,53,54,55,56,57,58,59,60}.…”

Section: 3mentioning

confidence: 99%

Two generalizations of homogeneity in groups with applications to regular semigroups

Araújo

Cameron

2015

Trans. Amer. Math. Soc.

110

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Let X be a finite set such that |X| = n and let i j n. A group G S n is said to be (i, j)-homogeneous if for every I, J ⊆ X, such that |I| = i and |J| = j, there exists g ∈ G such that Ig ⊆ J. (Clearly (i, i)-homogeneity is i-homogeneity in the usual sense.)A group G S n is said to have the k-universal transversal property if given any set I ⊆ X (with |I| = k) and any partition P of X into k blocks, there exists g ∈ G such that Ig is a section for P . (That is, the orbit of each k-subset of X contains a section for each k-partition of X.)In this paper we classify the groups with the k-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the (k−1, k)-homogeneous groups (for 2 < k ⌊ n+1 2 ⌋). As a corollary of the classification we prove that a (k − 1, k)-homogeneous group is also (k − 2, k − 1)-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the k-universal transversal property have the (k − 1)-universal transversal property.A corollary of all the previous results is a classification of the groups that together with any rank k transformation on X generate a regular semigroup (for 1 k ⌊ n+1 2 ⌋). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.

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“…The inclusion G W is said to be normal if M = j M (j) . Since our definition of the component M (j) is equivalent to the one given in [BPS07], an inclusion G W is normal if and only if the natural cartesian decomposition E of ∆ ℓ is M-normal, as defined in [BPS07]. The inclusion G W is normal if and only if for all T i there is a unique j such that T i M (j) ; that is, each T i acts trivially on all but one coordinate of ∆ ℓ .…”

Section: Normal Inclusionsmentioning

confidence: 99%

Inclusions of innately transitive groups into wreath products in product action with applications to 2-arc-transitive graphs

Praeger

Schneider

2016

Journal of Pure and Applied Algebra

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Abstract. We study (G, 2)-arc-transitive graphs for innately transitive permutation groups G such that G can be embedded into a wreath product Sym Γ wr S ℓ acting in product action on Γ ℓ . We find two such connected graphs: the first is Sylvester's double six graph with 36 vertices, while the second is a graph with 120 2 vertices whose automorphism group is Aut Sp(4, 4). We prove that under certain conditions no more such graphs exist.

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Quasiprimitive groups and blow-up decompositions

Cited by 6 publications

References 15 publications

Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

Two generalizations of homogeneity in groups with applications to regular semigroups

Inclusions of innately transitive groups into wreath products in product action with applications to 2-arc-transitive graphs

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