A note on a theorem of Dwyer and Wilkerson (Q5929489)

Let \(G\) be an elementary abelian \(p\)-group and \(X\) a finite \(G\)-CW complex. Over the finite field \(\mathbb F_p\), \textit{W. G. Dwyer} and \textit{C. W. Wilkerson} [Ann. Math. (2) 127, No. 1, 191-198 (1988; Zbl 0675.55011)] showed that the localization map induces an isomorphism \({H^*}_G (X^K) \to \mathcal U(S^{-1}{H^*}_G(X^K))\) of \(\mathcal A_p\)-modules, where \(K\) is a subgroup of \(G\), \(\mathcal U(S^{-1}{H^*}_G(X^K))\) the unstable classes of the localization \(S^{-1}{H^*}_G(X^K))\) and \(\mathcal A_p\) the \(\mathbb F_p\)-algebra generated by mod \(p\) Steenrod operations. In this paper, the author obtains similar isomorphism results for the non-elementary abelian group \(G_n=\mathbb Z_2 \times \mathbb Z_{2^n}\), \(n\geq 2\).