Sequences over finite fields defined by OGS and BN-pair decompositions of PSL2(q) recursively

Abstract

Factorization of groups into Zappa-Szep product, or more generally into k-fold Zappa-Szep product of its subgroups, is an interesting problem, since it eases the multiplication of two elements in a group, and has recently been applied for public-key cryptography as well. We give a generalization of the k-fold Zappa-Szep product of cyclic groups, which we call OGS decomposition. It is easy to see that existence of an OGS decomposition for all the composition factors of a non-abelian group G implies the existence of an OGS for G itself. Since the composition factors of a soluble group are cyclic groups, it obviously has an OGS decomposition. Therefore, the question of the existence of an OGS decomposition is interesting for non-soluble groups. The Jordan-Holder Theorem motivates us to consider an existence of an OGS decomposition for the finite simple groups. In 1993, Holt and Rowley showed that PSL_{2}(q) and PSL_{3}(q) can be expressed as a product of cyclic groups. In this paper, we consider an OGS decomposition of PSL_{2}(q) from a point of view different than that of Holt and Rowley. We look at its connection to the BN-pair decomposition of the group. This connection leads to sequences over F_{q}, which can be defined recursively, with very interesting properties, and which are closely connected to the Dickson and to the Chebyshev polynomials. Since every finite simple group of Lie-type has $BN-pair$ decomposition, the ideas of the paper might be generalized to further simple groups of Lie-type.