Modular invariants and the mod \(p\) cohomology algebra of the infinite symmetric group (Q1373080)

Let \(S_m\) be the symmetric group in \(m\) elements, \(\mathbb{F}_p\) be the prime field of order \(p\) and \(S_\infty\) be the infinite symmetric group. In this paper, using invariant theory, the author proves that \(H^*(S_\infty;\mathbb{F}_p)=\varprojlim_m H^*(S_m;\mathbb{F}_p)\). He first introduces the Dickson elements by means of the Dickson-Mùi invariants of \(\text{GL}(n,\mathbb{Z}_p)\) and then proves that the Hopf algebra \(H_*(S_\infty,\mathbb{F}_p)\) is freely generated by these Dickson elements, where the comultiplication is the diagonal map \[ \Delta\colon H_*(S_\infty;\mathbb{F}_p)\to H_*(S_\infty;\mathbb{F}_p)\otimes H_*(S_\infty;\mathbb{F}_p), \] induced by the diagonals \(S_m\to S_m\oplus S_m\). Dualizing, he obtains the above result.