A spread is a set of lines of PG(n,q), which partition the point set. A parallelism is a partition of the set of lines by spreads. A regulus of PG(3,q) is a set R of q+1 mutually skew lines such that any line intersecting three elements of R intersects all elements of R. A spread S of PG(3, q) is regular if for every three distinct elements of S, the unique regulus determined by them is a subset of S. A parallelism is regular if all its spreads are regular.
Regular parallelisms in PG(3,q) are known for any q ≡ 2 (mod 3) due to T. Penttila and B. Williams, 1998. In PG(3,5) these families of regular parallelisms are presented by two regular parallelisms with automorphisms of order 31 obtained early by A. Prince. Whether these are all regular parallelisms in PG(3,5) is an open question.
There are four relevant groups of order 5 and in the present paper it is established that the regular parallelisms of PG(3,5) cannot posses any authomorphism of order 5.