Group actions on spheres with rank one prime power isotropy

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We show that a rank two finite group G admits a finite G-CWcomplex X Sn with rank one prime power isotropy if and only if G does not p-involve Qd(p) for any odd prime p. This follows from a more general theorem which allows us to construct a finite G-CW-complex by gluing together a given G-invariant family of representations defined on the Sylow subgroups of G. © by International Press of Boston, Inc. All rights reserved.