(Oberseminar Geometrie, Gruppen- und Modelltheorie) Abstract: We investigate the action of the automorphism group of the free group on n generators Aut(Fn) on the set of generating n-tuples of a given (n-generated) group G. This action is related to the Product Replacement Algorithm, which is a practical algorithm for generating random elements in finite groups. A long standing conjecture of Wiegold states that when G is a finite simple group and n>2, this action is transitive. This conjecture is currently known only in very limited cases. Moreover, if the action is transitive then results of Gilman ('77) and Evans ('93) imply that Out(Fn) acts on the Aut(G)-equivalence classes of the generating tuples as a full symmetric or alternating group. Analogously, for certain infinite simple groups, known as Tarski monster groups, a recent joint work with Yair Glasner shows that when n>3, the Out(Fn) action is highly transitive, that is, k-transitive for any integer k.