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 finitep-groups coclassr, specially on periodicity results on these graphs. In particular, the structure of the subgraph inducedby ‘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 aperiodic parent. This shows that the periodicity results available inthe current literature for primes p≡5 mod 6 do not hold for the primes p≡1 mod 6. We also improve a known periodicity result in a special case of skeleton groups.