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}

Abstract

In this work we compute the induced transfer map: $$\bar\tau^\ast: \func{Im}(res^\ast:H^\ast(G) \to H^\ast(V)) \to \func{Im}(res^\ast: H^\ast (\Sigma_{p^n}) \to H^\ast(V))$$ in $\func{mod}p$-cohomology. Here $\Sigma_{p^{n}}$ is the symmetric group acting on an $n$-dimensional $\mathbb F_p$ vector space $V$, $G=\Sigma_{p^{n},p}$ a $p$-Sylow subgroup, $\Sigma_{p^{n-1}}\int \Sigma_{p}$, or $\Sigma_{p}\int \Sigma_{p^{n-1}}$. Some answers are given by natural invariants which are related to certain parabolic subgroups. We also compute a free module basis for certain rings of invariants over the classical Dickson algebra. This provides a computation of the image of the appropriate restriction map. Finally, if $ \xi :\func{Im}(res^\ast:H^\ast(G) \to H^\ast(V)) \to \func{Im}(res^\ast}: H^\ast(\Sigma_{p^n}) \to H^\ast(V)) $ is the natural epimorphism, then we prove that $\bar\tau^\ast=\xi$ in the ideal generated by the top Dickson algebra generator.