Costoya Ramos, María CristinaGomes, RafaelViruel Arbaizar, Antonio Ángel2024-09-272024-09-272024-05-22Results Math 79, 169 (2024)1422-6383http://hdl.handle.net/10347/34927We raise the question of the realizability of permutation mod ules in the context of Kahn’s realizability problem for abstract groups and the G-Moore space problem. Specifically, given a finite group G, we con sider a collection {Mi}ni=1 of finitely generated ZG-modules that admit asubmodule decomposition on which G acts by permuting the summands. Then we prove the existence of connected finite spaces X that realize each Mi as its i-th homology, G as its group of self-homotopy equiva lences E(X), and the action of G on each Mi as the action of E(X) onHi(X; Z)engAtribución 4.0 Internacional© 2024 The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International Licensehttp://creativecommons.org/licenses/by/4.0/Permutation modulesAutomorphismsGraphsPosetsHomotopy equivalencesjournal article10.1007/s00025-024-02199-z1420-9012open access