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 \(\{M_i\}_{i=1}^n\) of finitely generated \(\mathbb {Z}G\)-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 \(M_i\) as its i-th homology, G as its group of self-homotopy equivalences \(\mathcal {E}(X)\), and the action of G on each \(M_i\) as the action of \(\mathcal {E}(X)\) on \(H_i(X; \mathbb {Z})\).