Cochain sequences and the Quillen category of a coclass family

TL;DR

This paper introduces infinite cochain sequences and develops their homological algebra, demonstrating their utility in constructing coclass families and proving the equivalence of Quillen categories among most groups in these families.

Contribution

It presents a new framework of infinite cochain sequences and applies it to simplify coclass family constructions and establish Quillen category equivalences.

Findings

Infinite cochain sequences facilitate coclass family construction.

Most groups in a coclass family have equivalent Quillen categories.

Examples of infinite p-group families with equivalent Quillen categories.

Abstract

We introduce the concept of an infinite cochain sequence and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) and how they apply in proving that almost all groups in such a family have equivalent Quillen categories. We also include some examples of infinite families of p-groups from different coclass families that have equivalent Quillen categories.