Infinite sequences of $p$-groups with fixed coclass

TL;DR

This paper provides a detailed construction and explicit presentations of infinite sequences of finite p-groups with fixed coclass, expanding on prior sketches and demonstrating their utility in p-group theory.

Contribution

It offers a detailed construction method and explicit presentations for infinite p-group sequences with fixed coclass, including specific cases for primes 2 and 3.

Findings

Explicit presentations for sequences with primes (2,1), (2,2), and (3,1)

Enhanced understanding of the structure of infinite p-group sequences

Validation of the usefulness of these sequences in finite p-group theory

Abstract

Eick & Leedham-Green sketched a construction for infinite sequences of finite $p$-groups with fixed coclass. These infinite sequences have turned out to be very useful in the theory of finite $p$-groups. We exhibit a detailed description for the construction of the finite sequences and we determine presentations for the infinite sequences for the primes and coclasses $(2,1)$, $(2,2)$ and $(3,1)$.