The transfer in mod-\(p\) group cohomology between \(\Sigma_p\int\Sigma_{p^{n-1}}\), \(\Sigma_{p^{n-1}}\int\Sigma_p\) and \(\Sigma_{p^n}\). (Q847591)

Let \(V\) be a \(n\)-dimensional \(\mathbb F_p\) vector space and \(\Sigma_{p^n}\) the permutations on \(V\). Let \(H\) be a subgroup of the finite group \(G\). Then the Weyl group \(W_G(H)\) acts on the group cohomology \(H^*(H)\) and the inclusion map \(H\hookrightarrow G\) induces the restriction map \((\text{res}^G_H)^*\colon H^*(G)\to H^*(H)^{W_G(H)}\). In this paper the author computes the induced transfer map: \[ \overline\tau^*\colon\text{Im}(\text{res}^*\colon H^*(G)\to H^*(V))\to\text{Im}(\text{res}^*\colon H^*(\Sigma_{p^n})\to H^*(V)) \] in \(\text{mod\,}p\)-cohomology. Here \(G=\Sigma_{p^n,p}\) a \(p\)-Sylow subgroup, \(\Sigma_{p^{n-1}}\wr\Sigma_p\), or \(\Sigma_p\wr\Sigma_{p^{n-1}}\) where \(\wr\) denotes the wreath product. Certain rings of invariants which are related to parabolic subgroups have been studied. The author also computes a free module basis for certain rings of invariants over the Dickson algebra \(\mathbb F_p[y_1,\dots,y_n]^{\text{GL}(n,\mathbb F_p)}\). The method strongly depends on the action of Steenrod algebra on the rings of invariants.