On large sets of projective planes of orders 3 and 4

In 1850, Cayley (1850) [4] proved by a brief argument that a large set of Fano planes does not exist. In 1978, Magliveras conjectured that a large set of projective planes of order n will exist for all n≥3, provided that n is the order of a projective plane. In 1983, Chouinard II (1983) [5] constructed such large sets for n=3 by prescribing an automorphism of order 11 which acts semiregularly on the set of 55 planes. In this article, we construct new large sets of projective planes of order 3 by prescribing an automorphism of order 13. We classify all such large sets and determine their full automorphism groups. Moreover, we prove that the full automorphism group of any large set of projective planes of order 3, which is not isomorphic to the ones constructed thus far, is of order at most 5. Finally, in our effort to construct a large set of projective planes of order 4, which would consist of 969 planes, we construct 912 mutually disjoint projective planes of order 4 by prescribing C199 as an automorphism group.