Kourovka Notebook Problem 15.30

Definitions and mathematical statement

Let $p$ be a prime. A finite $p$-group is a finite group of order $p^n$ for some integer $n\geq 0$. For a group $G$, its Schur multiplier is the abelian group $M(G)=H_2(G,{\mathbb{Z}})$, where the coefficients have trivial $G$-action.

Is the following statement true? For every prime $p$ and every finite abelian $p$-group $A$, there exists a nonabelian finite $p$-group $G$ such that \[ M(G)\cong A. \] Here the prime is the same for $A$ and $G$, and nonabelian means that there exist $x,y\in G$ with $xy\ne yx$. Under the usual convention above, the trivial group is included among the possible groups $A$; the supplied statement gives no exception for it.

Short English statement

Must every finite abelian $p$-group occur, up to isomorphism, as the Schur multiplier of some nonabelian finite group whose order is a power of the same prime $p$?

Sources

• Local UnsolvedMath registry: permanent ID $701118$, exact source problem code catalogue entry, title Kourovka Notebook Problem 15.30. The supplied registry labels the problem as open; this status has not been independently verified.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), Problem 15.30. Proposed by Ya. G. Berkovich, according to the supplied record.
• The supplied background also cites the 21st-issue edition (2026), arXiv:1401.0300v48. The registry describes the collection as covering Issues 1--21 (1965--2026), edited by E. I. Khukhro and V. D. Mazurov. These bibliographic details are supplied attribution and have not been independently checked.