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

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 (Formula presented.) decomposition. It is easy to see that the existence of an (Formula presented.) decomposition for all the composition factors of a non-abelian group G implies the existence of an (Formula presented.) for G itself. Since the composition factors of a soluble group are cyclic groups, it has an (Formula presented.) decomposition. Therefore, the question of the existence of an (Formula presented.) decomposition is interesting for non-soluble groups. The Jordan–Hölder theorem motivates us to consider the existence of an (Formula presented.) decomposition for finite simple groups. In 1993, Holt and Rowley showed that (Formula presented.) and (Formula presented.) can be expressed as a product of cyclic groups. In this paper, we consider an (Formula presented.) decomposition of (Formula presented.) from a different point of view to that of Holt and Rowley. We look at its connection to the (Formula presented.) - (Formula presented.) decomposition of the group. This connection leads to sequences over (Formula presented.), 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 (Formula presented.) - (Formula presented.) decomposition, the ideas in this paper might be generalized to further simple Lie-type groups.