Sequences over Finite Fields Defined by OGS and BN-Pair Decompositions of PSL 2 ( 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 O G S decomposition. It is easy to see that the existence of an O G S decomposition for all the composition factors of a non-abelian group G implies the existence of an O G S for G itself. Since the composition factors of a soluble group are cyclic groups, it has an O G S decomposition. Therefore, the question of the existence of an O G S decomposition is interesting for non-soluble groups. The Jordan–Hölder theorem motivates us to consider the existence of an O G S decomposition for finite simple groups. In 1993, Holt and Rowley showed that P S L 2 ( q ) and P S L 3 ( q ) can be expressed as a product of cyclic groups. In this paper, we consider an O G S decomposition of P S L 2 ( q ) from a different point of view to that of Holt and Rowley. We look at its connection to the B N - p a i r decomposition of the group. This connection leads to sequences over F q , 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 B N - p a i r decomposition, the ideas in this paper might be generalized to further simple Lie-type groups.