Types of spreads and duality of the parallelisms of PG(3, 5) with automorphisms of order 13

Topalova, Svetlana; Zhelezova, Stela

doi:10.1007/s10623-018-0558-2

Cited by 12 publications

(5 citation statements)

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“…When dealing with several parallelisms at the same time we add some subscript or superscript to the symbols and S. The seminal book [18] covers the literature about parallelisms up to the year 2010. For the state of the art, various applications, connections with other areas of geometry and historical remarks, we refer also to [1], [3], [7], [13], [20], [26], [32] and the references therein.…”

Section: Basic Notions and Resultsmentioning

confidence: 99%

Clifford-like parallelisms

2018

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In this paper we focus on the description of the automorphism group Γ of a Clifford-like parallelism on a 3-dimensional projective double space P(H F ), ℓ , r over a quaternion skew field H (with centre a field F of any characteristic). We compare Γ with the automorphism group Γ ℓ of the left parallelism ℓ , which is strictly related to Aut(H). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose in such a way that Γ is either properly contained in Γ ℓ or coincides with Γ ℓ even though ℓ .Mathematics Subject Classification (2010): 51A15, 51J15

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Section: Basic Notions and Resultsmentioning

confidence: 99%

Clifford-like parallelisms

2018

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“…To learn more topics about parallelisms of PG(n, q), the reader is referred to [5,12,14,15,20,23,25].…”

Section: Discussionmentioning

confidence: 99%

The chromatic index of finite projective spaces

Xu¹,

Feng²

2022

Preprint

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A coloring of PG(n, q), the n-dimensional projective geometry over GF(q), is an assignment of colors to all lines of PG(n, q) so that any two lines with the same color do not intersect. The chromatic index of PG(n, q), denoted by χ ′ (PG(n, q)), is the least number of colors for which a coloring of PG(n, q) exists. This paper translates the problem of determining the chromatic index of PG(n, q) to the problem of examining the existences of PG(3, q) and PG(4, q) with certain properties. As applications, it is shown that for any odd integer n and q ∈ {3, 4, 8}, χ ′ (PG(n, q)) = n 1 q , which implies the existence of a parallelism of PG(n, q) for any odd integer n and q ∈ {3, 4, 8}.

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“…The number of reguli in the other spreads of PG(3, 7) is smaller. To recognize the type of the spreads in the constructed parallelisms, we use invariants based on the intersection of the spread with the reguli of the projective space as in [40]. These invariants can be represented by the triple (s, r, w), where s is the number of whole reguli in the spread and r (w) is the number of reguli which share exactly seven (six) lines (out of all q + 1 = 8 lines) with the spread.…”

Section: Uniform Deficiency One Parallelisms Of Pg(3 7)mentioning

confidence: 99%

Transitive Deficiency One Parallelisms of PG(3, 7)

Topalova

Zhelezova

2023

Mathematics

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Consider the n-dimensional projective space PG(n,q) over a finite field with q elements. A spread in PG(n,q) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A deficiency one parallelism is a partial parallelism with one spread less than the parallelism. A transitive deficiency one parallelism corresponds to a parallelism possessing an automorphism group which fixes one spread and is transitive on the remaining spreads. Such parallelisms have been considered in many papers. As a result, an infinite family of transitive deficiency one parallelisms of PG(n,q) has been constructed for odd q, and it has been proved that the deficiency spread of a transitive deficiency one parallelism must be regular, and its automorphism group should contain an elation subgroup of order q2. In the present paper we construct parallelisms of PG(3,7) invariant under an elation group of order 49 with some additional properties, and thus we succeed to obtain all (46) transitive deficiency one parallelisms of PG(3,7). The three parallelisms from the known infinite family are among them. As a by-product, we also construct a much bigger number (55,022) of parallelisms which have the same spread structure, but are not transitive deficiency one.

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Types of spreads and duality of the parallelisms of PG(3, 5) with automorphisms of order 13

Cited by 12 publications

References 32 publications

Clifford-like parallelisms

Clifford-like parallelisms

The chromatic index of finite projective spaces

Transitive Deficiency One Parallelisms of PG(3, 7)

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