Triple System And Fano Plane Structure In Zn

ABSTRACT

A triple system is an absolutely fascinating concept in projective geometry. This project is an extension of previously done work on triple systems, specifically the triples that fit into a Fano plane and the (i, j, k) triples of the quaternion group. Here, we have explored and determined the existence of triple systems in Z ∗ n for n = p, n = pq, n = 2mp and n = pqr with m ∈ N, p, q, r ∈ P, and p > q > r. A triple system in Z ∗ n has been denoted by (k1, k2, k3) where there exists ki > 1, i = 1, 2, 3, such that k 2 i ≡ 1(mod n) with k1k2 ≡ k3(mod n), k1k3 ≡ k2 (mod n) and k2k3 ≡ k1 (mod n). We have successfully proved that there exists no triples in Z ∗ n , for n = p and n = 2p, p ∈ P. Further, we have established the existence of triples in Z ∗ n , for n = pq, n = 2mp and n = pqr, where m ∈ N, p, q, r are odd primes and p > q > r. Finally, we have fitted the triples of Z ∗ n , n = 2mp and n = pqr into Fano Planes.