We explore transformation groups of manifolds of the form M x Sn, where M is an asymmetric manifold, that is, a manifold which does not admit any nontrivial action of a finite group. In particular, we prove that for n = 2, there exists an infinite family of distinct nondiagonal effective circle actions on such products. A similar result holds for actions of cyclic groups of prime order. We also discuss free circle actions on M x S1, where M belongs to the class of "almost asymmetric" manifolds considered previously by Puppe and Kreck.
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