Sequences over Finite Fields Defined by OGS and BN-Pair Decompositions of PSL2(q) Connected to Dickson and Chebyshev Polynomials

Abstract

The factorization of groups into a Zappa–Szép product, or more generally into a k-fold Zappa–Szép product of its subgroups, is an interesting problem, since it eases the multiplication of two elements in a group and has recently been applied to public-key cryptography. We provide a generalization of the k-fold Zappa–Szép product of cyclic groups, which we call OGS decomposition. It is easy to see that the 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 has an OGS decomposition. Therefore, the question of the existence of an OGS decomposition is interesting for non-soluble groups. The Jordan–Hölder theorem motivates us to consider the existence of an OGS decomposition for finite simple groups. In 1993, Holt and Rowley showed that PSL2(q) and PSL3(q) can be expressed as a product of cyclic groups. In this paper, we consider an OGS decomposition of PSL2(q) from a different point of view to that of Holt and Rowley. We look at its connection to the BN-pair decomposition of the group. This connection leads to sequences over Fq, which can be defined recursively, with very interesting properties, and are closely connected to Dickson and Chebyshev polynomials. Since every finite simple Lie-type group exhibits BN-pair decomposition, the ideas in this paper might be generalized to further simple Lie-type groups.