On transitive parallelisms of $$PG(3,4)$$ P G ( 3 , 4 )

Abstract

A parallelism in $$PG(n,q)$$ is transitive if it has an automorphism group which is transitive on the spreads. A parallelism is regular if all its spreads are regular. In $$PG(3,4)$$ no examples of transitive and no regular parallelisms are known. Transitive parallelisms in $$PG(3,4)$$ must have automorphisms of order 7. That is why we construct all 482 parallelisms with automorphisms of order 7 and establish that there are neither transitive, nor regular ones among them. We conclude that there are no transitive parallelisms in $$PG(3,4)$$ . The investigation is computer-aided. We use GAP (Groups, Algorithms, Programming—a System for Computational Discrete Algebra) to find a subgroup of order 7 and its normalizer in the automorphism group of $$PG(3,4)$$ . For all the other constructions and tests we use our own software written in C++.