Orbit isomorphic skeleton groups

Abstract

Recent development in the classification of \(p\)-groups often concentrate on the coclass graph \(\mathcal{G}(p,r)\) 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 \(\mathcal{G}(p,1)\). We find a family of skeleton groups in \(\mathcal{G}(7,1)\) whose \(6\)-step parent is not a periodic parent. This shows that the periodicity results available in the current literature for primes \(p\equiv 5\bmod 6\) do not hold for the primes \(p\equiv 1\bmod 6\). We also improve a known periodicity result in a special case of skeleton groups.