Finitep-groups of maximal class with ‘large’ automorphism groups

Abstract

AbstractThe classification ofp-groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the coclass graph𝒢${\mathcal{G}}$associated with thep-groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph𝒢∗${{\mathcal{G}}^{\ast}}$of𝒢${\mathcal{G}}$associated with thosep-groups of maximal class whose automorphism group orders are divisible byp-1${p-1}$. We describe the broad structure of𝒢∗${{\mathcal{G}}^{\ast}}$by determining its so-called skeleton. We investigate the smallest interesting casep=7${p=7}$in more detail using computational tools, and propose an explicit version of Conjecture W for𝒢∗${{\mathcal{G}}^{\ast}}$for arbitraryp≥7${p\geq 7}$. Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.