On the classification of groups of prime-power order by coclass: the 3-groups of coclass 2. (Q2852584)

This paper is part of an ongoing project aiming to classify finite \(p\)-groups via coclass. A finite \(p\)-group, \(G\), has coclass \(r=n-c\) if \(G\) has order \(p^n\) and nilpotency class \(c\). A useful tool in this investigation in the coclass graph, \(\mathcal G(p,r)\). The vertices of this graph are given by \(p\)-groups of coclass \(r\), one for each isomorphism type, and two vertices \(P\) and \(Q\) are joined if \(Q\cong P/L_c(P)\), where \(L_c(P)\) denotes the last non-trivial term of the lower central series of \(P\). Knowledge about this graph, and, in particular, the identification of a finite subgraph and finitely many operations which yield the whole graph from this given subgraph aids the classification project. Such a graph and operations have been given for \(\mathcal G(2,r)\) [\textit{B. Eick} and \textit{C. Leedham-Green}, Bull. Lond. Math. Soc. 40, No. 2, 274-288 (2008; Zbl 1168.20007)], and \(p\)-groups of maximal class (that is coclass 1) are well understood.NEWLINENEWLINE In this highly technical paper the authors investigate \(\mathcal G(3,2)\). They identify the skeleton graph, which `although sparse, exhibits the broad nature of \(\mathcal G(3,2)\)'. The authors investigate this skeleton graph and suggest a finite subgraph of it will be the required finite graph. Finally the authors conclude by stating a new conjecture, Conjecture W, `about the graph-theoretic operations needed to describe \(\mathcal G(p,r)\) for arbitrary \(p\) and \(r\)'.