3-groups are not determined by their integral cohomology rings

TL;DR

This paper computes the integral cohomology rings of certain 3-groups and shows that for groups of order 3^n with n ≥ 5, different groups can have identical cohomology rings, indicating non-uniqueness.

Contribution

It demonstrates that the integral cohomology ring does not uniquely determine 3-groups for orders 3^n with n ≥ 5, providing explicit examples.

Findings

Identified pairs of non-isomorphic 3-groups with isomorphic cohomology rings for n ≥ 5

Computed the integral cohomology rings of a specific family of 3-groups

Showed that cohomology rings are not complete invariants for 3-groups

Abstract

We compute the integral cohomology rings of a family of 3-groups. As a corollary, we exhibit, for each n greater than or equal to 5, a pair of groups of order 3^n whose integral cohomology rings are isomorphic.