4-Homogeneous groups

Kantor, William M.

doi:10.1007/bf01111290

Cited by 15 publications

(4 citation statements)

References 3 publications

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“…where q is a prime power congruent to 3 (mod 4), or G = AGL(1, 8), AΓL (1,8) or AΓL (1,32), with n = 8, 8, 32 respectively (Kantor [24]); • if G is 4-homogeneous but not 4-transitive, then G = PGL(2, 8), PΓL (2,8) or PΓL (2,32), with n = 9, 9, 33 respectively (Kantor [23]); • using the Classification of Finite Simple Groups, all the t-transitive groups for t ≥ 2 have been determined; lists are available in [11,15]. In particular, the only 6-transitive groups are the symmetric and alternating groups, and the only 4-or 5-transitive groups are these and the Mathieu groups M 11 , M 12 , M 23 and M 24 .…”

Section: Homogeneity and Transitivitymentioning

confidence: 99%

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The classification of partition homogeneous groups with applications to semigroup theory

2016

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Let λ = (λ 1 , λ 2 , . . .) be a partition of n, a sequence of positive integers in non-increasing order with sum n. Let Ω := {1, . . . , n}. An ordered partition P = (A 1 , A 2 , . . .) of Ω has type λ if |A i | = λ i .Following Martin and Sagan, we say that G is λ-transitive if, for any two ordered partitions P = (A 1 , A 2 , . . .) and Q = (B 1 , B 2 , . . .) of Ω of type λ, there exists g ∈ G with A i g = B i for all i. A group G is said to be λhomogeneous if, given two ordered partitions P and Q as above, inducing the setsThe first goal of this paper is to classify the λ-homogeneous groups (Theorems 1.1 and 1.2). The second goal is to apply this classification to a problem in semigroup theory.Let Tn and Sn denote the transformation monoid and the symmetric group on Ω, respectively. Fix a group H Sn. Given a non-invertible transformation a ∈ Tn \ Sn and a group G Sn, we say that (a, G) is an H-pair if the semigroups generated by {a} ∪ H and {a} ∪ G contain the same non-units, that is, a, G \ G = a, H \ H. Using the classification of the λ-homogeneous groups we classify all the Sn-pairs (Theorem 1.7). For a multitude of transformation semigroups this theorem immediately implies a description of their automorphisms, congruences, generators and other relevant properties (Theorem 8.5).This topic involves both group theory and semigroup theory; we have attempted to include enough exposition to make the paper self-contained for researchers in both areas.The paper finishes with a number of open problems on permutation and linear groups.

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Section: Homogeneity and Transitivitymentioning

confidence: 99%

“…Clearly a t-transitive group is thomogeneous. Livingstone and Wagner [32] showed that the converse is true if 5 ≤ t ≤ n/2; the t-homogeneous, but not t-transitive groups for t = 2, 3, 4 were determined by Kantor [23,24].…”

Section: Introductionmentioning

confidence: 99%

The classification of partition homogeneous groups with applications to semigroup theory

2016

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“…Note. The result stated by KA~TO~ in [8] is proved there only under the additional assumption that GA[ A has order ~: 8, which is quite sufficient for our purposes.…”

Section: Characterization Of Families Of Rank 3 Permutation Groups Bymentioning

confidence: 89%

Characterization of families of rank 3 permutation groups by the subdegrees I

Higman

1970

Arch. Math

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“…(We may assume without loss that t |V |/2. Without the classification it can be shown that such a group is, with certain known exceptions, always t-transitive, see [5][6][7]; and the list of the t-transitive groups, which can be found in [1], follows from the classification.) For our more general concept, the determination of orbit-t-homogeneous groups is not complete, but we give a number of results in this direction.…”

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confidence: 99%

Orbit-Homogeneity in Permutation Groups

Cameron

Dent

2006

Bull. London Math. Soc.

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This paper introduces the concept of orbit-homogeneity of permutation groups: a group G is orbitt-homogeneous if two sets of cardinality t lie in the same orbit of G whenever their intersections with each G-orbit have the same cardinality. For transitive groups, this coincides with the usual notion of t-homogeneity. This concept is also compatible with the idea of partition transitivity introduced by Martin and Sagan.Further, this paper shows that any group generated by orbit-t-homogeneous subgroups is orbitt-homogeneous, and that the condition becomes stronger as t increases up to n/2 , where n is the degree. So any group G has a unique maximal orbit-t-homogeneous subgroup Ωt(G), and Ωt(G) Ω t−1 (G). Some structural results for orbit-t-homogeneous groups, and a number of examples, are also given.

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4-Homogeneous groups

Cited by 15 publications

References 3 publications

The classification of partition homogeneous groups with applications to semigroup theory

The classification of partition homogeneous groups with applications to semigroup theory

Characterization of families of rank 3 permutation groups by the subdegrees I

Orbit-Homogeneity in Permutation Groups

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