3-groups are not determined by their integral cohomology rings

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.

On The Homotopy Type of 𝐵𝐺 for Certain Finite 2-Groups 𝐺

“…A similar fact is known for various metacyclic 2-groups, among which one counts the quaternion and semidihedral groups [15]. In a recent paper, I. Leary exhibited examples of distinct 3-groups having isomorphic integral cohomology rings [12]. Further examples of distinct finite 2-groups, which mod-2 cohomology algebra cannot tell apart, are given in [16].…”

The Modular Isomorphism Problem: a survey

“…For instance for all groups of order 32 the cohomology rings were computed by Rusin [Rus89] and some turned out to be isomorphic, though (MIP) had been solved over the field of 2 elements for groups of order 32 already at the time [Mak76]. Later Leary found 3-groups of order 3 n , for n ≥ 5, which have isomorphic cohomology rings over the integers and hence over any ring [Lea95]. According to the comment at the end of [SS96a] these groups were the original motivation for Salim and Sandling to study (MIP) for groups of order p 5 .…”

p -Groups are Not Determined by Their Integral Cohomology Groups

“…Any elements A ′ and B ′ generating G such that A ′ commutes with G ′ and B ′ has order p must satisfy a similar presentation, except that the new q may be the old one multiplied by the nth power of any integer coprime to p. It follows that for fixed m there are n isomorphism types of such groups. For n = 2 these are the groups already considered by the author and Yagita [5,8].…”

“…The only previous examples of p-groups having isomorphic integral cohomology groups known to the author are a family of (pairs of) p-groups for p at least 5 discovered by Yagita [8] and a similar family for p = 3, for which the author has been able to show further that the groups have isomorphic integral cohomology rings [5]. The advantages of this paper are that a wider class of examples is exhibited, including p-groups for p = 2, and that very little calculation is involved.…”