Kourovka Notebook Problem 21.137

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite $p$-group $G$ distinguish the set $P=\{g^p:g\in G\}$ from the subgroup $G^p$ generated by that set. A finite $p$-group $H$ is powerful if $[H,H]\leq H^p$ for odd $p$, or $[H,H]\leq H^4$ for $p=2$. The exponent is the least positive integer killing every element.

Statement recorded in the supplied source.

If $P$ is a subgroup, must it be powerful? In particular, (a) for odd $p$ and $G$ of exponent $p^2$, must $P$ be abelian; and (b) for a $2$-group $G$ of exponent $8$ whose squares form a subgroup, must that subgroup be abelian? [S1][S2]

Short English statement

Determine whether a power image that is already a subgroup must be powerful, including the proposed abelianness conclusions for exponent p squared and exponent eight.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2646 (KOU-21.137). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 E. I. Khukhro and V. D. Mazurov (editors), The Kourovka Notebook, Unsolved Problems in Group Theory, No.21, version48 (6 October2026), Problem 21.137 and its remarks. https://arxiv.org/abs/1401.0300v48.