Kourovka Notebook Problem 15.97

Definitions and mathematical statement

Fix a prime $p$. A group $G$ is locally finite if every finitely generated subgroup of $G$ is finite. A $p$-element is an element of order $p^k$ for some integer $k\geq 0$. A $p'$-group here means a periodic group in which every element has order relatively prime to $p$.

Following the definition attributed in the source to S. N. Chernikov, $G$ satisfies the $p$-minimal condition if there is no infinite strictly descending chain \[ H_1>H_2>H_3>\cdots \] of subgroups of $G$ such that, for every $i\geq 1$, the set $H_i\setminus H_{i+1}$ contains a $p$-element.

For a prime $q$, the quasicyclic group $C_{q^\infty}$ is the union of cyclic groups of orders $q,q^2,q^3,\ldots$ under the usual inclusions. A group is Chernikov if it has a subgroup of finite index isomorphic to a finite direct product \[ \prod_{j=1}^{m} C_{q_j^\infty}, \qquad m\geq 0, \] where the $q_j$ are primes. The empty product is permitted, so finite groups are included.

Under the finite-length convention, a subnormal series is a chain \[ 1=G_0\trianglelefteq G_1\trianglelefteq\cdots\trianglelefteq G_r=G, \qquad r\geq 0, \] where each $G_{i-1}$ is normal in $G_i$. The supplied source does not explicitly state whether its term "subnormal series" permits transfinite series; the displayed formulation uses the finite-length convention, and any intended broader interpretation remains unresolved.

Question. For every prime $p$ and every locally finite group $G$ satisfying the $p$-minimal condition and admitting such a subnormal series with each factor $G_i/G_{i-1}$ finite or a $p'$-group, must \[ P_p(G):=\langle x\in G:\operatorname{ord}(x)=p^k\text{ for some integer }k\geq 0\rangle \] be a Chernikov subgroup of $G$? The conclusion concerns the subgroup generated by all $p$-elements; it does not additionally require that this subgroup be a $p$-group.

Short English statement

Does the subgroup generated by all $p$-elements of a locally finite group have to be Chernikov when the group satisfies the specified descending-chain condition and has a subnormal series with factors that are finite or $p'$-groups? The draft explicitly uses a finite subnormal series; whether the source intends a broader series convention requires review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701185$, exact source problem code catalogue entry, title Kourovka Notebook Problem 15.97. The supplied registry status is open; this label is not an independently verified literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 15.97, attributed to N. S. Chernikov, proposed in 2002. The supplied background identifies the 15th issue (2002) and the 21st-issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently checked.
• The source credits S. N. Chernikov with the definition of the $p$-minimal condition; this attribution is distinct from the problem's attribution to N. S. Chernikov.