Orbit isomorphic skeleton groups

Abstract

Recent development in the classification of p-groups often concentrate on the coclass graph G(p,r) associated with the finite p-groups coclass r, specially on periodicity results on these graphs. In particular, the structure of the subgraph induced by `skeleton groups' is of notable interest. Given their importance, in this paper, we investigate periodicity results of skeleton groups. Our results concentrate on the skeleton groups in G(p,1). We find a family of skeleton groups in G(7,1) whose 6-step parent is not a periodic parent. This shows that the periodicity results available in the current literature for primes p≡5mod6 do not hold for the primes p≡1mod6. We also improve a known periodicity result in a special case of skeleton groups.