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The investigation of subgroups of the general linear group GL(n,q), the group of non-singular n x n matrices over a finite field F of order q, has been an actively studied area in computational group theory in recent years. Since the order of GL(n,q) grows exponentially, even very basic questions might cause serious computational problems and dealing with matrix groups becomes tricky. Despite these difficulties several algorithms have been developed which efficiently explore properties of a subgroup G \leq GL(n,q). In 1997 A. Niemeyer and C. E. Praeger presented an algorithm to determine whether or not G contains both the special linear group SL(n,q) of all matrices of determinant 1 and various classical groups. For this purpose certain elements in G are sought by making random selections of elements from G. These so called primitive prime divisor elements, or ppd-elements, are very likely to occur in classical groups. Thus, under reasonable conditions, with a high probability finding two such elements allows to conclude that SL(n,q), or another classical group respectively, is contained in G. However the algorithm requires G to act irreducibly on the underlying vector space V.

In this thesis after introducing all necessary background information we focus on the question whether we could ignore this additional irreducibility condition assuming that finding two ppd-elements already implies V being G-irreducible. We give a lower bound for the proportion of pairs (g,h) of two ppd-elements where this implication holds. More precisely we determine a constant c such that for all q with n sufficiently large the mentioned proportion is bigger than 1-c/q^{n-1}.