Abstract
There is exactly one compact 1-dimensional Lie group having 27 components and nilpotence class three. We give a presentation for the integral cohomology ring of (the classifying space of) this group. We show that the groups of order 3 4 can be distinguished by their first few integral cohomology groups, and exhibit a pair of groups of order 3 5 having isomorphic integral cohomology rings.
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