The Postnikov tower and the Steenrod problem (Q2706601)

A Moore space is a topological space whose reduced homology vanishes in all dimensions except one. If \(G\) is a finite group and \(M\) is a finitely generated \(ZG\)-module, then \(M\) is a Steenrod representation if there exists a Moore space \(X\) with \(G\)-action such that the homology of \(X\) is isomorphic to \(M\) as a \(ZG\)-module. In [Can. J. Math. 29, 421-428 (1977; Zbl 0364.55015)] \textit{J. E. Arnold jun.} proved that if \(G\) is cyclic, then every finitely generated \(ZG\)-module is a Steenrod representation. Also in [K-Theory 1, 325-335 (1987; Zbl 0659.55008)] \textit{P. Vogel} proved that for any \(ZG\)-module \(M\) there is a \(G\)-Moore space \(X\) which realizes \(M\), then \(G\) has only cyclic subgroups. In the paper under review, it is proved that \(M\) is a Steenrod representation as a \(ZG\)-module if and only if \(M\) is a Steenrod representation as a \(ZH\)-module for all \(p\)-Sylow subgroups \(H\) of \(G\), for all primes \(p \mid |G|\).