On finite $p$-groups satisfying given laws

TL;DR

This paper characterizes certain properties of finite p-groups within group varieties, establishing conditions under which these groups are bounded in nilpotency class or coclass, based on the laws they satisfy.

Contribution

It provides new criteria linking group laws to bounds on nilpotency class and coclass in finite p-groups within varieties.

Findings

Bound on nilpotency classes in powerful p-groups within a variety

Finiteness of finite p-groups of a given coclass in a variety

Characterization of group laws determining these properties

Abstract

A variety of groups does not contain all metabelian groups if and only if there is an absolute bound for the nilpotency classes of powerful $p$-groups in the given variety. Similarly, a variety contains only finitely many finite $p$-groups of any given coclass if and only if not every group that is an extension of an abelian group by an elementary abelian $p$-group belongs to that variety.