Transitive Deficiency One Parallelisms of PG(3, 7)

Autor: Svetlana Topalova, Stela Zhelezova
Jazyk: angličtina
Rok vydání: 2023
Předmět:
Zdroj: Mathematics, Vol 11, Iss 11, p 2458 (2023)
Druh dokumentu: article
ISSN: 2227-7390
DOI: 10.3390/math11112458
Popis: 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.
Databáze: Directory of Open Access Journals
Nepřihlášeným uživatelům se plný text nezobrazuje