Triple System and Fano Plane Structure in Z∗n
Gikunda, Dennis Kinoti
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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 2 N, p; q; r 2 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 ki2 ≡ 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 2 P. Further, we have established the existence of triples in Z∗ n, for n = pq, n = 2mp and n = pqr, where m 2 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.