The image of the transfer map

Theorem 1.1 Suppose that G is a finite group, and k is a field of characteristic p. Let H be a collection of subgroups of G. Denote by K the collection of subgroups K of G with the property that the Sylow p-subgroups of the centralizer CG(K) are not conjugate to a subgroup of any of the groups in H. Let J be the ideal in H∗(G, k) given by the sum of the images of transfer from subgroups in H, and J ′ be the ideal in H∗(G, k) given by the intersection of the kernels of restriction to subgroups in K. Then the ideals J and J ′ have the same radical, √ J = √ J ′.