The cyclic parallelism of PG(3,5) (Q1266377)

A spread of the projective space \(P = \text{PG}(3,q)\) is a set of \(q^2+1\) pairwise non-intersecting lines of \(P\). A parallelism of \(P\) is a family of \(q^2+q+1\) spreads of \(P\) which partition the set of lines of \(P\). A parallelism is called cyclic, if there is a collineation of \(P\) of order \(q^2+q+1\) that permutes its spreads in a cyclic way. A parallelism is called regular, if any of its spreads is regular. Up to now, no example of a regular parallelism of \(P\) was known. In the paper under review the author determines (by the use of a computer) all cyclic parallelisms of \(\text{PG} (3,5)\). Up to projective equivalence, there are 45 cyclic parallelisms of \(\text{PG} (3,5)\), two of them being regular.