Plotkin’s dimension-series problems for groups

Determine whether, for every group G, the equality D_ω(G)=γ_ω(G) holds and whether, for every positive integer c, there exists an integer m=m(c) such that D_m(G)=1 for every nilpotent group G of class at most c.

Background

The paper recalls two problems formulated by Plotkin in 1973. They concern the relationship between the intersection of the dimension subgroups and the intersection of the lower central subgroups, and the existence of a uniform vanishing index for dimension subgroups of nilpotent groups of bounded class. The paper states that these group-theoretic problems remain unresolved, while it proves the corresponding statements for metabelian Lie rings.

References

In 1973, Plotkin formulated two problems that remain open. For a group $G$, put

D_\omega(G)=\bigcap_{n\geq1}D_n(G), \qquad \gamma_\omega(G)=\bigcap_{n\geq1}\gamma_n(G).

The problems ask: \begin{enumerate}[(i)] \item Is $D_\omega(G)=\gamma_\omega(G)$ for every group $G$? \item For every positive integer $c$, is there an integer $m=m(c)$ such that $D_m(G)=1$ for every nilpotent group $G$ of class at most $c$? \end{enumerate}

— Dimension subrings of Lie rings  (2610.02042 - Ionin et al., 1 Oct 2026) in Section 1, subsection “Plotkin's problems for metabelian Lie rings”

The corresponding question for groups remains open: is $D_5(G)\subseteq\gamma_4(G)$ for every group $G$? SeeProblem~20.84.

— Dimension subrings of Lie rings  (2610.02042 - Ionin et al., 1 Oct 2026) in Section 1, subsection “Plotkin's problems for metabelian Lie rings”