Nilpotent associative algebras and coclass theory

We initiate the investigation of nilpotent associative algebras using the coclass as primary invariant. We consider the coclass graph GF(r)GF(r) associated with the nilpotent associative FF-algebras of coclass r  . First, we describe the infinite paths of GF(r)GF(r) via a precise structure description of the inverse limits corresponding to these infinite paths. Then we prove that the number of essentially different infinite paths in GF(r)GF(r) is finite if and only if |F||F| is finite or r≤1r≤1. Thus the results of this paper state and prove equivalents to the Coclass Conjectures C and D as introduced by Leedham-Green and Newman for the coclass theory of finite p-groups.