  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.
