Kourovka Notebook Problem 14.91

Definitions and mathematical statement

Let $p$ be a fixed prime. A finite $p$-group is a finite group $G$ with $|G|=p^n$ for some integer $n\geq 0$.

The supplied question asks whether, for every $a\geq 0$, there exists a finite $p$-group of abundance $a$. Schematically, this is \[ \forall a\geq 0\quad \exists\,G\text{ a finite }p\text{-group}\quad \operatorname{abundance}(G)=a. \] Here $\operatorname{abundance}(G)$ is only a placeholder for the source's undefined term "abundance", not a defined invariant. The supplied record neither defines abundance nor specifies the domain of $a$. Consequently, this display records the requested quantifier structure but is not a fully specified mathematical statement. Both the definition of abundance and the intended range of $a$ must be supplied before the question can be made precise.

Short English statement

For each fixed prime $p$, the question asks whether every nonnegative admissible value $a$ occurs as the abundance of a finite $p$-group. The supplied record leaves both abundance and the domain of $a$ undefined.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701075$, exact source problem code catalogue entry, titled "Kourovka Notebook Problem 14.91".
• The record attributes the problem to G. Fernández--Alcober, proposed in 1999, and cites The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), Problem 14.91; it also identifies the 21st issue edition (2026), arXiv:1401.0300v48.
• The supplied collection description identifies E. I. Khukhro and V. D. Mazurov as editors. The registry labels the problem "open"; this is a supplied classification, not an independently verified mathematical or literature assessment. The cited source has not been independently consulted for this draft.