Group actions on spheres with rank one prime power isotropy

We show that a rank two finite group $G$ admits a finite $G$-$\mathrm{CW}$-complex $X \simeq S_n$ with rank one prime power isotropy if and only if $G$ does not $p^{\prime}$-involve $\mathrm{Qd}(p)$ for any odd prime $p$. This follows from a more general theorem which allows us to construct a finite $G$-$\mathrm{CW}$-complex by gluing together a given $G$-invariant family of representations defined on the Sylow subgroups of $G$.

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Research partially supported by NSERC Discovery Grant A4000. The second author was also partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) through the research program BİDEB-2219.

Received 6 April 2015

Accepted 23 November 2015

Published 24 July 2017