Let p be an odd prime. For a compact Lie group G and an elementary abelian p-group A of G, one may define the Weyl group \(W_A\) of A in a similar fashion as defining the Weyl group of a maximal torus, such that \(W_A\) acts on \(H^*(BA;R)\) for any coefficient ring R, and the image of the restriction \(H^*(BG;R)\rightarrow H^*(BA;R)\) lies in \(H^*(BA;R)^{W_A},\) the sub-algebra of \(H^*(BA:R)\) of \(W_A\) -invariant elements. In this paper, we consider the projective unitary group \(PU(p^{m})\) and one of its maximal elementary abelian p-subgroup \(A_m,\) of which the Weyl group is isomorphic to \(Sp_{2{m}}({\mathbb {F}}_p).\) Then the theory of \(Sp_{2{m}}({\mathbb {F}}_p)\) -invariant polynomials over \({\mathbb {F}}_p\) may be applied to study the cohomology of \(BPU(p^{m}),\) the classifying space of \(PU(p^{m}).\) Following a theorem by Quillen, we deduce several theorems on \(H^*(BPU(p^{m});{\mathbb {F}}_p)\) modulo the nilradical from results on invariant polynomials.
