A result concerning the existence of certain finite Minkowski - 2 - Structures

Quattrocchi, Pasquale; Fiori, Carla

doi:10.1007/bf01918525

Cited by 4 publications

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“…Assume d = 4. If |X | ≥ 7 then we have |X | = 11 by the result in [16] and Proposition 7 yields that G is a copy of the Mathieu group of degree 11. If |X | = 6 then G = Alt(X ) by Proposition 6 in [17].…”

Section: Higher Degreesmentioning

confidence: 91%

“…We have thus the following result. [16] we have |X | − 1 = 11, whence |X | = 12 and Proposition 13 yields that G is a copy of the Mathieu group of degree 12. If |X | = 7 then G = Alt(X ) by Proposition 6 in [17].…”

Section: Higher Degreesmentioning

confidence: 93%

“…Assume |X | ≥ 7 and let G be a sharply 4-transitive permutation set on X containing the identity and having the property that each permutation j ∈ G exchanging two elements and fixing three further elements of X is necessarily an involution. It was proved in [16] that under these assumptions we must have |X | = 11. It was also remarked that the above property is certainly satisfied if G, besides containing the identity, is such that g ∈ G always implies g −1 ∈ G, that is G is an invertible permutation set.…”

Section: Introductionmentioning

confidence: 99%

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Bonisoli

Quattrocchi

2000

Journal of Algebraic Combinatorics

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All known finite sharply 4-transitive permutation sets containing the identity are groups, namely S 4 , S 5 , A 6 and the Mathieu group of degree 11. We prove that a sharply 4-transitive permutation set on 11 elements containing the identity must necessarily be the Mathieu group of degree 11. The proof uses direct counting arguments. It is based on a combinatorial property of the involutions in the Mathieu group of degree 11 (which is established here) and on the uniqueness of the Minkowski planes of order 9 (which had been established before): the validity of both facts relies on computer calculations. A permutation set is said to be invertible if it contains the identity and if whenever it contains a permutation it also contains its inverse. In the geometric structure arising from an invertible permutation set at least one block-symmetry is an automorphism. The above result has the following consequences. i) A sharply 5-transitive permutation set on 12 elements containing the identity is necessarily the Mathieu group of degree 12. ii) There exists no sharply 6-transitive permutation set on 13 elements. For d ≥ 6 there exists no invertible sharply d-transitive permutation set on a finite set with at least d + 3 elements. iii) A finite invertible sharply d-transitive permutation set with d ≥ 4 is necessarily a group, that is either a symmetric group, an alternating group, the Mathieu group of degree 11 or the Mathieu group of degree 12.

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Section: Higher Degreesmentioning

confidence: 91%

Section: Higher Degreesmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Bonisoli

Quattrocchi

2000

Journal of Algebraic Combinatorics

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“…From each sharply k-transitive permutation set G, k >/2, we can define an incidence structure; the large amount of papers on this subject shows that many results have been obtained through the study of the geometrical properties of the incidence structure associated to G (see, for example, [3], [4], [7], [8], [13]- [16]). Interesting examples of permutation sets have been constructed in [10] and [12].…”

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confidence: 99%

Over the non-existence of sharply 3-transitive permutation sets containing sharply 2-transitive permutation subsets

Quattrocchi

1987

Geom Dedicata

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Invertible Sharplyn-Transitive Sets

Polster

1998

Journal of Combinatorial Theory, Series A

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We prove various results about sharply n-transitive sets of homeomorphisms of`n ice'' topological spaces like the real line and the circle. Our main results concern sharply 3-transitive sets G of homeomorphisms of the circle to itself. If G=G &1 , then G contains a real hyperbolic part (a set of involutions of the circle having special properties). We show that every real hyperbolic part is the hyperbolic part of a real abstract oval in the sense of Buekenhout and that every real abstract oval arises from a topological oval in a flat projective plane. This establishes a new relationship between flat Minkowski planes that admit automorphisms which are circle-symmetries and flat projective planes containing topological ovals. We also consider sharply n-transitive sets of permutations acting on finite sets. We find that our results about flat geometries and sharply n-transitive sets of homeomorphisms have counterparts in the finite case.

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A result concerning the existence of certain finite Minkowski - 2 - Structures

Cited by 4 publications

References 1 publication

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Over the non-existence of sharply 3-transitive permutation sets containing sharply 2-transitive permutation subsets

Invertible Sharplyn-Transitive Sets

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