An odd order group without the dimension property

TL;DR

This paper constructs a specific finitely presented group demonstrating a counterexample to the dimension property in finite 3-groups, challenging previous results in the field.

Contribution

It provides the first explicit example of a finite 3-group lacking the dimension property, disproving earlier assumptions.

Findings

Existence of a finite 3-group without the dimension property

Counterexample to previous theoretical results

Identification of an element of order 3 in the dimension quotient

Abstract

We construct a finitely presented group $G$ such that the $7$th dimension quotient $G\cap(1+\varpi(\mathbb Z G)^7)/\gamma_7(G)$ has an element of order $3$; this immediately leads to a finite $3$-group without the dimension property. This contradicts a series of results by N. Gupta.