Triple System and Fano Plane Structure in Z n
- September 5, 2020
- Posted by: RSIS Team
- Categories: IJRIAS, Mathematics
International Journal of Research and Innovation in Social Science (IJRISS) | Volume V, Issue IV, April 2020 | ISSN 2454–6186
Triple System and Fano Plane Structure in Z n
Dennis Kinoti Gikunda, Benard Kivunge
Kenyatta University, Kenya
Abstract: – A triple system is an absolutely fascinating concept in projective geometry. This paper 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 for n = p, n = pq and n = 2mp with m εN, p, qεℙ, and p > q, where N is the set of natural numbers, ℙ is the set of primes and is the set of units in Zn. A triple system in 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 also investigated the number of triples in and determined the general formula for getting the triples. Further, we have fitted the triples into Fano planes and established the projective geometry structure for the above defined .
AMS Subject Classification: 20B05
Key Words: Geometry, Triple Systems, Fano Planes, Prime Numbers, Units.
Triple System and Fano Plane Structure in
