Let X denote the group of all automorphisms of a finite Lebesgue space equipped with the weak topology. For T E A, let UrT denote its maximal spectral type. Theorem 1. There is a dense G3 subset G c X such that, for each T E G and k(1), . k(l) E Z+, k'(1), ... , k'(l') E Z+, the convolutions UTk(1) * * UTk(U) and a TkI(1) *... * UTkO(1) are mutually singular, provided that (k(1) . k(l)) is not a rearrangement of (k'(1), ... , k'(l')). Theorem 1 has the following consequence. Theorem 2. X has a dense G3 subset F c G such that for T E F the following holds: For any k: N -. Z {O} and 1 E Z {O}, the only way that TI, or any factor thereof can sit as a factor in T k(l) x T k(2) X *.* is inside the ith coordinate c-algebra for some i with k(i) = 1 . Theorem 2 has applications to the construction of certain counterexamples, in particular nondisjoint automorphisms having no common factors and weakly isomorphic automorphisms that are not isomorphic.