The origins of coclass theory (Q6139347)

If \(G\) is a finite group of order \(p^{n}\) (\(p\) a prime) and nilpotency class \(c\), then the coclass of \(G\) is defined as \(n-c\). The coclass invariant was introduced by \textit{C. R. Leedham-Green} and \textit{M. F. Newman} in [Arch. Math. 35, 193--202 (1980; Zbl 0437.20016)] for the purpose of study and partially classifying the \(p\)-groups. Groups of coclass \(1\) (that is of maximal class) have been studied extensively in the literature and their intricate structure was one of the main incentives for Leedham-Green and Newman to generalise maximal class to coclass. The paper under review is a survey which is intended to briefly explain the main results of Leedham-Green and Newman, to describe the developments that led to five coclass conjectures (see the paper) and to outline the ongoing impact on \(p\)-group theory.