Kourovka Notebook Problem 14.36

Definitions and mathematical statement

A subgroup $H\leq G$ is subnormal in $G$ if there are an integer $n\geq 0$ and subgroups \[ H=H_0\trianglelefteq H_1\trianglelefteq\cdots\trianglelefteq H_n=G. \] A group $G$ is a $T$-group if every subnormal subgroup of $G$ is normal in $G$. It is an $\overline{T}$-group if every subgroup $K\leq G$, including $G$ itself, is a $T$-group; thus, for all $H\leq K\leq G$, subnormality of $H$ in $K$ implies $H\trianglelefteq K$.

A group $G$ is locally graded if every nontrivial finitely generated subgroup $K\leq G$ has a proper subgroup $L<K$ of finite index, so that $1<[K:L]<\infty$. A group is periodic if every element has finite order; hence $G$ is non-periodic if it contains an element of infinite order.

The question is whether, for every group $G$, \[ \left(G\text{ is non-periodic, locally graded, and an }\overline{T}\text{-group}\right) \quad\Longrightarrow\quad \forall x,y\in G,\;xy=yx. \] The supplied registry labels this question open; that status is supplied source metadata and has not been independently verified.

Short English statement

Must every locally graded group containing an element of infinite order be abelian if every subgroup is a $T$-group, meaning that its subnormal subgroups are normal in it?

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701021$, exact source problem code catalogue entry, title Kourovka Notebook Problem 14.36. The supplied status is open.
• F. de Giovanni, proposer, 1999. The supplied attribution identifies The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), original problem number 14.36, 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 verified.