Kourovka Notebook Problem 5.55

Definitions and mathematical statement

For a prime $p$, a finite $p$-group is a group of order $p^n$ for some integer $n\geq 0$. Interpret the source's term "multiplicator" in its standard sense as the Schur multiplier $M(H)=H_2(H,{\mathbb{Z}})$. An embedding is an injective group homomorphism.

Find a finite $p$-group $G$ such that, for every finite $p$-group $H$, \[ M(H)=0\quad\Longrightarrow\quad \text{there is no injective homomorphism }G\longrightarrow H. \] The same prime $p$ is used for $G$ and every candidate ambient group $H$. The source does not specify whether finding an example for some prime suffices or whether examples are requested for every prime; this quantifier qualification remains unresolved.

The source additionally states that every finite group can be embedded in a group with trivial multiplicator. This is retained as a supplied background assertion; its ambient group is not explicitly required to be finite or a $p$-group.

Short English statement

Find a finite $p$-group that embeds into no finite $p$-group with trivial Schur multiplier. The source leaves the intended quantification over primes unspecified. Its background embedding assertion imposes no explicit finiteness or $p$-group condition on the ambient group.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700336$, exact source code catalogue entry, title "Kourovka Notebook Problem 5.55." The registry labels the problem open; this is a supplied status, not an independently verified mathematical or literature assessment.
• Attributed to J. Wiegold, proposed in 1976. The supplied reference is The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), Problem 5.55, as included in the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.