Groups of prime-power order and coclass theory

• Applicant: Professorin Dr. Bettina Eick
• Subject Area: Mathematics
• Term: from 2017 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 386837064

• Final Report Year: 2022

Final Report Abstract

This project investigated the graph G(p, 1) associated with the p-groups of maximal class (or, equivalently, of coclass 1). The vertices of this graph correspond one-to-one to the isomorphism types of p-groups of maximal class and there is an edge G → H if G ∼ = H/λ(H), where λ(H) is the last non-trivial term of the lower central series of H. Each graph G(p, 1) is an infinite graph. Investigating this graph is one way of working towards a classification of the p-groups of maximal class. The graphs G(2, 1) and G(3, 1) are well-understood due to work by Blackburn. The graph G(5, 1) is experimentally well-understood. The graphs G(p, 1) for p ≥ 7 have been investigated by Leedham-Green and McKay, but many details about their structure are still widely unknown. These graphs are the main focus of this project. Conjecture W by Eick, Leedham-Green, Newman and O’Brien suggests that each graph G(p, 1) can be constructed from a finite subgraph using two types of periodic patterns: one of these is the pattern proved independently by du Sautoy and by Eick and Leedham-Green, and the other is a major open problem in coclass theory. This project introduced Galois trees as a new tool to investigate G(p, 1). Galois trees are subtrees of G(p, 1) consisting of groups whose automorphism group has a specific form. In the first part of this project, the Galois trees in G(p, 1) for some small primes p have been investigated experimentally. A result of this investigation was a conjectural description of the important features of Galois trees. In the second part of this project many of the conjectured features of Galois trees have been proved in a joint work by Cant, Dietrich, Eick and Moede. Additionally, for the primes p ≡ 5 mod 6, it has been shown that Galois trees can be applied to obtain further insights into a possible approach towards proving Conjecture W . In summary, this project can be considered as a first explicit and constructive description on a second periodic pattern in the graphs G(p, 1) for primes p ≥ 7 with p ≡ 5 mod 6. Among the side-results of this project are new software tools that can be used to analyse the graphs G(p, r) experimentally. The software tools have been developed in packages for the computer algebra system GAP and they are freely available.

Publications

• Classifying nilpotent associative algebras: small coclass and finite fields. In ’Algorithmic and experimental methods in algebra, geometry, and number theory’ (Springer, Cham), 213–229 (2017)
  Bettina Eick and Tobias Moede
  
• Coclass theory for finite nilpotent associative algebras. Exp. Math. 26, 267–274 (2017)
  Bettina Eick and Tobias Moede
  
• Finite p-groups of maximal class with ’large’ automorphism groups. J. Group Theory 20, 227–256 (2017)
  Heiko Dietrich and Bettina Eick
  
• Polynomials describing the multiplication in finitely generated torsion-free nilpotent groups. J. Symbolic Comput. 92, 203–210 (2019)
  Alexander Cant and Bettina Eick
  
• Galois trees in the graph associated with the p-groups of maximal class
  Alexander Cant, Heiko Dietrich, Bettina Eick and Tobias Moede