For a based space \(X\) and a prime \(p\), let \({\mathcal E}(X)\) denote the group of based self-homotopy equivalences on \(X\), and let \({\mathcal E}_{\#}^m(X)\) (resp. \({\mathcal E}_{\#p}(X)\)) be its subgroup given by \({\mathcal E}_{\#}^m(X)=\text{Ker }[{\mathcal E}(X)\to \prod_{i\leq m}\text{aut }\pi_i(X)]\) (resp. \({\mathcal E}_{\#p}(X)= \text{Ker }[{\mathcal E}(X)\to \prod_{i\leq N}\text{aut }\pi_i(X;\mathbb Z/p)]\), where \(N=\dim X\) denotes the topological dimension or homotopical dimension). For a local coefficient system \({\mathcal M}=\{{\mathcal M}_x\}_{x\in X}\), let \({\mathcal E}(X,{\mathcal M})\) denote the subgroup of \({\mathcal E}(X)\) which induces automorphisms on \({\mathcal M}\).
In this paper, the authors study a generalization of a result due to E. Dror and A. Zabrodsky [Topology 18, 187–197 (1979; Zbl 0417.55008)] concerning the nilpotency in homotopy equivalences, and they prove several generalizations of it. For example, they prove that if \(X\) is a finite Postnikov piece and \(G\) is a subgroup of \({\mathcal E}(X,{\mathcal M})\) which acts nilpotently on both \(\pi_*(X)\) and \({\mathcal M}\), then \(G\) acts nilpotently on \(H^*(X,{\mathcal M})\). Moreover, they also prove that if \(\pi_1(X)\) is a nilpotent group with subgroup \(G\subset {\mathcal E}(X)\) which acts nilpotently on \(\pi_{i\leq N}(X)_{(p)}\) for any prime \(p\) and \(0\) and the nilpotency orders of all these actions are bounded, then \(G\) is nilpotent. Their proof is based on several basic facts of group theory and careful investigations concerning the self homotopy equivalences of spaces with local coefficients.