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We prove that if a finite group $G$ acts smoothly on a manifold $M$ so thatall the isotropy subgroups are abelian groups with rank $\leq k$, then $G$ actsfreely and smoothly on $M \times \bbS^{n_1} \times...\times \bbS^{n_k}$ forsome positive integers $n_1,...n_k$. We construct these actions using arecursive method, introduced in an earlier paper, that involves abstract fusionsystems on finite groups. As another application of this method, we prove thatevery finite solvable group acts freely and smoothly on some product of sphereswith trivial action on homology.

Fusion systems and group actions with abelian isotropy subgroups