A note on skeleton groups in coclass graphs

Recent studies of p-groups of coclass r concentrate on the coclass graph (p,r). While the detailed structure of (p,r) is unknown, it is known that its general structure is dominated by the subgraph of 'skeleton groups'. The original definition of these groups is technical, but some modifications for special cases have been used successfully in the literature. Given their importance, in this paper we define and investigate skeleton groups more rigorously. In particular, we study their isomorphism problem, which is a crucial step towards understanding the skeleton subgraph of (p,r). During our work we identified erroneous arguments for constructing isomorphisms in a proof of the 2013 paper on (3, 2). We correct these errors here by proving the required results in a more general context.