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

TL;DR

This paper introduces the frame of the graph associated with p-groups of maximal class, providing a new approach to their classification by leveraging the Lazard correspondence and subgraph analysis.

Contribution

It defines the frame of the graph G(p) and demonstrates its utility in classifying p-groups of maximal class through the existence of specific normal subgroups.

Findings

Every p-group of maximal class has a normal subgroup of order at most p.

The quotient of such a subgroup lies within the frame of G(p).

The frame closely approximates the full graph, aiding classification.

Abstract

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.