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- Title
Transitive Deficiency One Parallelisms of PG(3, 7).
- Authors
Topalova, Svetlana; Zhelezova, Stela
- Abstract
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 q 2 . 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.
- Subjects
AUTOMORPHISM groups; POINT set theory; FINITE fields; PROJECTIVE spaces; PERMUTATION groups
- Publication
Mathematics (2227-7390), 2023, Vol 11, Issue 11, p2458
- ISSN
2227-7390
- Publication type
Article
- DOI
10.3390/math11112458