On transitive parallelisms of \(PG(3,4)\)
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, Programminga 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++.
transitivity parallelism automorphisms spread 11t71 51e23 05b05