p-Groups are not determined by their integral cohomology groups

TL;DR

This paper demonstrates that integral cohomology groups do not uniquely identify p-groups by providing examples of non-isomorphic p-groups with identical cohomology groups, using a method involving kernels of homomorphisms from Lie groups.

Contribution

It introduces a novel approach to constructing p-groups with identical cohomology groups, challenging the idea that cohomology determines group structure.

Findings

Identified pairs of p-groups with isomorphic integral cohomology groups

Developed a method using kernels of homomorphisms from Lie groups

Corrected previous examples of 2-groups with this property

Abstract

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).