Eilenberg-Mac Lane spectra as equivariant Thom spectra

We prove that the G-equivariant mod p Eilenberg-Mac Lane spectrum arises as an equivariant Thom spectrum for any finite, p-power cyclic group G, generalizing a result of Behrens and the second author in the case of the group C-2. We also establish a construction of HZ((p)), and prove intermediate results that may be of independent interest. Highlights include constraints on the Hurewicz images of equivariant spectra that admit norms, and an analysis of the extent to which the nonequivariant HFp arises as the Thom spectrum of a more than double loop map.