The frame of the graph associated with the p-groups of maximal class

The graph G(p) associated with the p-groups of maximal class is a major tool in their classification. We introduce a subgraph of the graph G(p) called its frame. Its construction is based on the Lazard correspondence. We show that every p-group of maximal class has a normal subgroup of order at most p whose quotient is in the frame. Since the frame is close to the full graph, it offers a new approach towards the classification of the p-groups of maximal class.

💡 Research Summary

The paper introduces a new subgraph of the well‑studied graph G(p) that encodes all p‑groups of maximal class. The authors call this subgraph the “frame”. While the classical picture of G(p) consists of an isolated vertex C_{p²}, an infinite mainline S₂→S₃→…, and a family of finite branches B_i attached to the mainline, the detailed structure of each branch becomes increasingly intricate for primes p ≥ 5. Earlier work by Leedham‑Green and McKay introduced “skeletons” – finite subtrees that capture much of the branch structure – but the construction required a normal subgroup N of order dividing p^{18}(p−1), which is far too large to give fine‑grained insight.

To overcome this limitation the authors construct the frame using p‑adic number theory and the Lazard correspondence. Let K=ℚ_p(θ) be the p‑adic field obtained by adjoining a primitive p‑th root of unity θ, and let 𝒪 be its ring of integers with the descending chain of ideals 𝔭_i (index p^i). For each i ≥ 0 they consider the set \hat H_i of P‑linear surjective homomorphisms γ: 𝔭_i∧𝔭_i → 𝔭_{2i+1}, where P=⟨θ⟩ is the cyclic group of order p. From a given γ they define an ideal J_i(γ) generated by the Jacobi‑type expressions γ(γ(x∧y)∧z)+…; this ideal is always of the form 𝔭_λ. The quotient L_{i,m}(γ)=𝔭_i/𝔭_m with Lie bracket