Kourovka Notebook Problem 20.84

Definitions and mathematical statement

For a group $G$, define its lower central series by $\gamma_1(G)=G$ and \[ \gamma_{n+1}(G)=\langle [x,y]:x\in\gamma_n(G),\ y\in G\rangle, \qquad [x,y]=x^{-1}y^{-1}xy. \] The group $G$ has nilpotency class exactly $3$ if $\gamma_4(G)=\{1\}$ and $\gamma_3(G)\ne\{1\}$.

Under the conventional integral interpretation of dimension subgroups, let ${\mathbb{Z}}[G]$ be the integral group ring, let \[ \varepsilon:{\mathbb{Z}}[G]\longrightarrow{\mathbb{Z}}, \qquad \varepsilon\!\left(\sum_{g\in G}a_g g\right)=\sum_{g\in G}a_g, \] where each sum has finite support, and let $I(G)=\ker\varepsilon$ be the augmentation ideal. Define \[ D_5^{{\mathbb{Z}}}(G)=\{g\in G:g-1\in I(G)^5\}, \] where $I(G)^5$ is the fifth ideal power.

With this interpretation, the question proposed by E. Rips is whether \[ \exists\text{ a group }G\quad \gamma_4(G)=\{1\},\qquad \gamma_3(G)\ne\{1\},\qquad D_5^{{\mathbb{Z}}}(G)\ne\{1\}. \] No finiteness or finite generation hypothesis is imposed in the supplied statement. The coefficient ring for the dimension subgroup is not explicitly specified there; the integral interpretation above requires confirmation against the original source.

Short English statement

Does there exist a group of nilpotency class exactly $3$ whose fifth dimension subgroup is nontrivial? Under the conventional integral interpretation, this means that some $g\ne1$ satisfies $g-1\in I(G)^5$ in ${\mathbb{Z}}[G]$. The supplied record does not explicitly specify the coefficient ring.

Sources

• Local UnsolvedMath registry, permanent ID $701742$, exact source problem code catalogue entry, titled Kourovka Notebook Problem 20.84. The supplied registry labels the problem open; this label has not been independently verified.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem $20.84$, proposed by E. Rips in $2022$. The supplied background identifies the 20th issue (2022) and the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied source text and bibliographic attribution were used without independent consultation of the cited editions.