Kourovka Notebook Problem 6.1

Definitions and mathematical statement

Interpret the source notation $C$ as the centralizer in the fixed ambient group $G$; this interpretation is implicit rather than explicitly defined in the supplied record. For a subset $S\subseteq G$, set \[ C_G(S)=\{g\in G:gs=sg\text{ for every }s\in S\},\qquad C_G^2(S)=C_G(C_G(S)). \] For $x\in G$, write $C_G^2(x)=C_G^2(\{x\})$. A subgroup $H\leq G$ is $C$-closed if $H=C_G^2(H)$, and is weakly $C$-closed if \[ \forall x\in H,\qquad C_G^2(x)\leq H. \] A group is locally finite if every subgroup generated by finitely many elements is finite.

Problem (V. A. Antonov and N. F. Sesekin, 1978). Describe, up to group isomorphism, all locally finite groups $G$ satisfying \[ \forall H\leq G,\qquad H\neq G\ \Longrightarrow\ \left(\forall x\in H,\ C_G(C_G(\{x\}))\leq H\right). \] The quantifier ranges over all proper subgroups, including the trivial subgroup when it is proper, and all centralizers are taken in $G$. Both finite and infinite locally finite groups are included.

The supplied background attributes to Gaschütz a study of finite groups whose proper subgroups are all $C$-closed. That background concerns the stronger closure condition and supplies no precise theorem or bibliographic reference for it.

Short English statement

Describe all locally finite groups in which every proper subgroup contains the ambient double centralizer of each of its elements. Here locally finite means that every finitely generated subgroup is finite. The centralizer meaning of $C$ is implicit in the supplied source.

Sources

• Local UnsolvedMath registry, permanent ID $700348$, exact source problem code catalogue entry, title Kourovka Notebook Problem 6.1; supplied complete source record, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 6.1, sixth issue (1978); the supplied record identifies the twenty-first issue edition (2026), arXiv:1401.0300v48. Proposed by V. A. Antonov and N. F. Sesekin. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The source background attributes the finite-group study for the $C$-closed condition to Gaschütz, without supplying a publication citation or a precise result.
• The registry labels the problem open. This is a supplied status label, not an independently verified literature assessment. No literature-verification date or explicit reference URL is supplied.