A short survey on Coclass Graphs

Leedham-Green & Newman [11] defined the coclass of a p-group of order pn and nilpotency class c as r = n−c. The investigation of the p-groups of a fixed coclass led to deep results in p-group theory (see the book of Leedham-Green & McKay), applications (see for example [1, 12]), and generalisations to other algebraic objects (see for example [5, 8]). In the last decade, the focus in coclass theory is on the investigation of the coclass graph G(p, r) associated with the finite p-groups of coclass r. It is conjectured that this infinite graph can be described by a finite subgraph and several “periodic patterns”. The aim of this talk is to give a survey on the known periodicity results, the outstanding problems, and a recent new result for the graph G(p, 1). Some details are given below.