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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}

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}

Nondas E. Kechagias

In this work we compute the induced transfer map:$$\bar{\tau}^{\ast }:{}{\Im}\left( res^{\ast }:H^{\ast }\left( G\right)\rightarrow H^{\ast }\left( V\right) \right) \rightarrow \Im\left(res^{\ast }:H^{\ast }\left( \Sigma _{p^{n}}\right) \rightarrow H^{\ast}\left( V\right) \right)$$in ${}{\mod}\ p$-cohomology. Here $\Sigma _{p^{n}}$\ is the symmetric groupacting 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 certainparabolic subgroups. We also compute a free module basis for certain ringsof invariants over the classical Dickson algebra. This provides acomputation of the image of the appropriate restriction map. Finally, if $%\xi :\Im\left( res^{\ast }:H^{\ast }\left( G\right) \rightarrowH^{\ast }\left( V\right) \right) \rightarrow \Im\left( res^{\ast}:H^{\ast }\left( \Sigma _{p^{n}}\right) \rightarrow H^{\ast }\left(V\right) \right) $ is the natural epimorphism, then we prove that $\bar{\tau}%^{\ast }=\xi $ in the ideal generated by the top Dickson algebra generator.\\

Journal of Homotopy and Related Structures, Vol. 4(2009), No. 1, pp. 153-179