  For each prime p, we exhibit pairs of p-groups all of whose integral
cohomology groups are isomorphic. The method used involves very little
calculation. The groups are exhibited as kernels of homomorphisms from a
compact Lie group G to U(1), and the main result is that kernels of `similar'
elements of Hom(G,U(1)) have isomorphic integral cohomology groups.
  The 2-groups constructed in this version have been corrected (there was a
mistake in the presentations given in the published paper).
