Kourovka Notebook Problem 14.90

Definitions and mathematical statement

For a group $P$, define its lower central series by \[ \gamma_1(P)=P,\qquad \gamma_{i+1}(P)=[\gamma_i(P),P]\quad(i\geq1), \] where $[H,P]$ is the subgroup generated by commutators $h^{-1}x^{-1}hx$ with $h\in H$ and $x\in P$. Define its upper central series by \[ \zeta_0(P)=1,\qquad \zeta_{j+1}(P)/\zeta_j(P)=Z(P/\zeta_j(P))\quad(j\geq0), \] where $Z(H)$ denotes the center of a group $H$. The nilpotency class $c$ is the least nonnegative integer for which $\gamma_{c+1}(P)=1$.

The source asks whether a finite $p$-group $P$ of "abundance" $a$ and nilpotency class $c$ admits an integer bound $t=t(a)$ such that \[ \gamma_i(G)=\zeta_{c-i+1}(G)\qquad(i\geq t). \] Here the source does not define abundance, introduces $G$ without identifying it with $P$, and gives no upper bound on $i$, although the upper central series index becomes negative when $i>c+1$.

Conditional reconstruction requiring review. If $G$ means $P$, and the intended range consists of the indices for which both displayed series terms are defined, the question becomes: for every admissible abundance value $a$, does there exist an integer $t(a)\geq1$, depending only on $a$, such that for every prime $p$ and every finite $p$-group $P$ of abundance $a$ and nilpotency class $c$, \[ \gamma_i(P)=\zeta_{c-i+1}(P) \qquad\text{for every integer }i\text{ with }t(a)\leq i\leq c+1? \] The endpoint $i=c+1$ gives the automatic equality $1=1$. The definition and admissible values of abundance remain unspecified, so this reconstruction is not a fully determined mathematical statement.

The supplied source states that the assertion holds for $a=0$ with $t=1$, because $P$ then has maximal class, and that for $a=1$ one can take $t=3$. These are preserved as source assertions, without independent verification.

Short English statement

The question asks whether abundance alone bounds how far down the lower central series a finite $p$-group must be before it agrees with the upper central series read in reverse. Precision requires a definition of abundance, clarification that $G=P$, and an intended index range. The supplied registry labels the problem open; that label is not an independently verified status assessment.

Sources

• Local UnsolvedMath registry, permanent ID $701074$, exact source problem code catalogue entry, title Kourovka Notebook Problem 14.90; supplied status: open. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 14.90, attributed to G. Fernández--Alcober, proposed in 1999; the supplied background identifies the 14th issue (1999) and the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The statements concerning abundance $0$ and $1$ are supplied with the problem text. No separate proofs or supporting references for these statements are supplied. The cited notebook has not been independently consulted for this draft.