Polynomial invariant rings isomorphic as modules over the steenrod algebra

The paper is concerned with rings of polynomial invariants of finite groups . In particular, it will be shown that these rings are isomorphic as module s over the Steenrod algebra P* if and only if the group representations are pointwise conjugate. An application to cohomology is the construction of c lassifying spaces of finite groups which are not homotopy equivalent, but w here the cohomology rings are isomorphic as unstable modules over the (topo logical) Steenrod algebra.