ABSTRACT
It is always of interest to determine the transitive groups that can act on a parallelism, and we consider such a study later on in this text. In this chapter, we consider parallelisms in PG(3; q) that admit a collineation of order a p-primitive divisor of q31, so we develop a more general study than merely the consideration of transitivity, noting that a group acting transitively on the parallelism has order divisible q
Certainly, the set of all transitive parallelisms is far from complete. For example, we do provide a new in…nite class of transitive parallelisms in the next chapter, but there are also a large number of non-regular but transitive parallelisms in PG(3; 5) because of Prince [176] (45 total parallelisms, of which 43 are not regular). Similarly, one might consider the classi…cation of 2-transitive parallelisms, which we have noted has been resolved by the author, where is shown that only the two regular parallelisms in PG(3; 2) are possible and these parallelisms admit a collineation group isomorphic to PSL(2; 7) acting 2-transitively on the 1 + 2 + 22 spreads. For work in PG(3; q), it is possible to utilize the quite old work of Mitchell [171] 1911 and Hartley [64] 1928. Using this material, we are able to obtain strong restrictions on the nature of transitive parallelisms, which, in particular, provides an alternative proof for the classi…cation of 2-transitive parallelisms, when q 6= 4 and furthermore shows that non-solvable groups admitting p-primitive elements have a very restrictive structure and involve only PSL(2; 7) or A7. In the transitive case, only PSL(2; 7) can occur and q = 2. For more general work on doubly transitive t-parallelisms or transitive t-parallelisms, a considerable …nite group theory is required. The work in this chapter follows the article by Biliotti, Jha, and Johnson [22], to a certain extent. We …rst are reminded that a ‘primitive group’is a transitive group
that does not admit a non-trivial ‘block’(there is not a non-trivial partition of the set upon which the group acts that permutes the elements of the partition). When the group is a linear group, the blocks tend to be vector subspaces that are permuted. In this chapter, primitive groups are discussed, but this is not where we start. We consider groups
acting on parallelisms of PG(3; q), especially when acting on the set of 1 + q + q2 = (q3 1)=(q 1) spreads and ask what can be said of a collineation group that contains an element g, whose order is a (prime) p-primitive divisor of (q3 1)? As we know a ‘p-primitive divisor’of pk 1 is a divisor that does not divide pj 1, for j < k. Now there is one situation, namely, 43 1, where prime p-primitive divisors do not exist, so our result will not apply for parallelisms of PG(3; 4). It turns out that this assumption on p-primitivity is extremely powerful, and we can say a great deal about the possible parallelisms.
