New Regular Parallelisms of PG(3, 5)

Abstract

We construct all regular parallelisms of PG(3,5) with automorphisms of order 3. Their number is 8. The two cyclic parallelisms found by Prince are among them. The other six ones are the first examples of noncyclic regular parallelisms and the first examples of regular parallelisms that do not belong to the infinite families of Penttila and Williams.

Conjugates for Finding the Automorphism Group and Isomorphism of Design Resolutions

“…The method is suitable for resolutions of designs with big automorphism groups. It was developed and to be useful in connection with problems concerning classification of parallelisms of P G(3, 4) and P G(3, 5) with predefined automorphisms [20], [19], [21], [22]. The advantages of the present method in these cases are based on the fact that the considered resolutions possess predefined automorphisms and this algorithm uses the predefined groups, while general purpose algorithms do not.…”

“…Since not many different types of parallelisms are known, computer-aided constructions are of particular interest. A great deal of the computer-aided investigations imply classifications of t-parallelisms with predefined automorphism groups [12], [13], [14], [15], [17], [18], [20], [19], [21], [22], [23]. In all these works the authors use the normalizer of the constructive automorphism group in order to filter away isomorphic parallelisms.…”

“…with automorphisms of orders 7 and 5 [20], [21] and of parallelisms of P G(3, 5) (|G D |=2 9 • 3 2 • 5 6 • 13 • 31) with automorphisms of orders 13 and 3 [19], [22].…”

“…Section 3 explains how the necessary set of conjugates constructed as shown in section 2 can be used to determine if a resolution R with given automorphisms is isomorphic to another resolution R ′ . The method was used to determine the pairs of duals of regular parallelisms of P G(3, 5) with automorphisms of order 3 [22].…”

Pencilled regular parallelisms

“…by specifying the ground field or by adding topological conditions. Recent contributions in this spirit are [2], [3], and [27]; see also the references at the end of Section 2. In the present article we are concerned with regular parallelisms, that is, parallelisms that are made from regular spreads.…”