Realization of Permutation Modules via Alexandroff Spaces

We raise the question of the realizability of permutation modules in the context of Kahn’s realizability problem for abstract groups and the G -Moore space problem. Specifically, given a finite group G , we consider a collection ᵢ\ᵢ₌₁ⁿ { M i } i = 1 n of finitely generated ZG Z G -modules that admit a submodule decomposition on which G acts by permuting the summands. Then we prove the existence of connected finite spaces X that realize each Mᵢ M i as its i -th homology, G as its group of self-homotopy equivalences E(X) E ( X ) , and the action of G on each Mᵢ M i as the action of E(X) E ( X ) on Hᵢ(X; Z) H i ( X ; Z ) .