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Factorization of groups into Zappa-Szep product, or more generally intok-fold Zappa-Szep product of its subgroups, is an interesting problem, since iteases the multiplication of two elements in a group, and has recently beenapplied for public-key cryptography as well. We give a generalization of thek-fold Zappa-Szep product of cyclic groups, which we call OGS decomposition. Itis easy to see that existence of an OGS decomposition for all the compositionfactors 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, itobviously has an OGS decomposition. Therefore, the question of the existence ofan OGS decomposition is interesting for non-soluble groups. The Jordan-HolderTheorem motivates us to consider an existence of an OGS decomposition for thefinite simple groups. In 1993, Holt and Rowley showed that PSL_{2}(q) andPSL_{3}(q) can be expressed as a product of cyclic groups. In this paper, weconsider an OGS decomposition of PSL_{2}(q) from a point of view different thanthat of Holt and Rowley. We look at its connection to the BN-pair decompositionof the group. This connection leads to sequences over F_{q}, which can bedefined recursively, with very interesting properties, and which are closelyconnected to the Dickson and to the Chebyshev polynomials. Since every finitesimple group of Lie-type has $BN-pair$ decomposition, the ideas of the papermight be generalized to further simple groups of Lie-type.

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