The isomorphism problem for graded algebras and its application to mod-p cohomology rings of small p-groups

TL;DR

This paper presents an algorithm to determine if two graded algebras are isomorphic and applies it to classify the mod-p cohomology rings of small p-groups, enhancing understanding of their algebraic structures.

Contribution

The paper introduces an effective algorithm for testing graded isomorphism of algebras and classifies cohomology rings of p-groups up to order 100.

Findings

Algorithm successfully distinguishes non-isomorphic cohomology rings.

Complete classification of cohomology rings for p-groups of order ≤ 100.

Provides new insights into the structure of mod-p cohomology rings.

Abstract

The mod-p cohomology ring of a non-trivial finite p-group is an infinite dimensional, finitely presented graded unital algebra over the field with p elements, with generators in positive degrees. We describe an effective algorithm to test if two such algebras are graded isomorphic. As application, we determine all graded isomorphisms between the mod-p cohomology rings of all p-groups of order at most 100.