Equivariant generating hypotheses for finite groups

We disprove Bohmann’s equivariant generating hypothesis for every nontrivial finite group $G$, even when all $\text{RO}(H)$-graded homotopy groups at every subgroup $H$ are tested. For each fixed $G$ and prime $p$ dividing $\mathrm{\mid }G\mathrm{\mid }$, we construct ghosts on finite $G$-spectra with arbitrarily long nonzero composition powers. We also prove that the homotopy-module functors are nonfull and construct non-equivalent finite $G$-spectra with isomorphic full homotopy modules. Our constructions use circle power maps and cyclic permutations of products of projective spaces, and are motivated by Ma–Xu’s categorical method in the motivic setting and the projective-space power maps.