Cochain sequences and the Quillen category of a coclass family

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.

Essential cohomology for elementary abelian p-groups

For an odd prime p the cohomology ring of an elementary abelian p-group is polynomial tensor exterior. We show that the ideal of essential classes is the Steenrod closure of the class generating the top exterior power. As a module over the polynomial algebra, the essential ideal is free on the set of Mui invariants.