Kourovka Notebook Problem 15.82

Definitions and mathematical statement

A group $G$ is periodic if every element of $G$ has finite order. A subgroup $U\leq G$ is a $2$-subgroup if every element of $U$ has order $2^n$ for some integer $n\geq 0$. A group is locally finite if each of its finitely generated subgroups is finite. The condition $U\trianglelefteq G$ means that $gUg^{-1}=U$ for every $g\in G$.

The source asks whether, for every periodic group $G$ and every $2$-subgroup $U\leq G$, \[ \left[U\text{ is strongly isolated in }G\right] \quad\Longrightarrow\quad \left[G\text{ is locally finite}\ \lor\ U\trianglelefteq G\right]. \] The supplied record does not define strongly isolated. This hypothesis is retained verbatim as an unresolved source term; the displayed implication requires the intended definition before it constitutes a fully specified mathematical statement.

Short English statement

Must a periodic group containing a strongly isolated $2$-subgroup either be locally finite or have that subgroup normal? The supplied record leaves the meaning of "strongly isolated" unspecified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701170$, exact source problem code catalogue entry, title Kourovka Notebook Problem 15.82. The record labels the problem open; this is supplied source metadata, not an independently verified status assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 15.82, 15th issue (2002); also identified in the supplied background with the 21st issue edition (2026), arXiv:1401.0300v48. Proposed by A. I. Sozutov and N. M. Suchkov. These bibliographic details are supplied attribution and have not been independently verified.