INVARIANTS OF THE SYMMETRIC GROUP

Let $R = k[y_1,\cdots,y_n] \otimes E[x_1, \cdots, x_n]$ with characteristic $k = p > 2$ (odd prime), where $$\mid$y_i$\mid$ = 2, $\mid$x_i$\mid$ = 1$ and $y_i = \betax_i, \beta$ is the Bockstein homomorphism. Topologically, $R = H^*(B(Z/p)^n,k)$. For a symmetric group $\sum_n, R^{\sum_n} = k[\sigma_1,\cdots,\sigma_n] \otimes E[d\sigma_1, \cdots, d\sigma_n]$ where d is the derivation satisfying $d(y_i) = x_i$ and $d(x_iy_i) = x_iy_i + x_jy_i, 1 \leq i, j \leq n$. We give a direct proof of this theorem by using induction.