Kourovka Notebook Problem 12.6

Definitions and mathematical statement

Let $p$ be a prime. Here a $p$-group means a group in which every element has finite order equal to a power of $p$, and a $p$-subgroup means a subgroup with this property. The source does not explicitly specify its convention for infinite $p$-groups; no finiteness, local finiteness, or bounded-exponent assumption is added here.

Let $G$ be a finitely generated group and let $H\leq G$ be a $p$-subgroup. Assume that \[ \forall N\trianglelefteq G,\qquad N\leq H\ \Longrightarrow\ N=\{1\}, \] and that \[ HX=XH\qquad\text{for every subgroup }X\leq G, \] where $HX=\{hx:h\in H,\ x\in X\}$ and $XH=\{xh:x\in X,\ h\in H\}$. Define the normal closure of $H$ in $G$ by \[ H^G=\left\langle g^{-1}hg:h\in H,\ g\in G\right\rangle, \] and its centralizer by \[ C_G(H^G)=\{g\in G:gy=yg\text{ for every }y\in H^G\}. \] Since $H^G$ is normal in $G$, its centralizer is normal in $G$, so the quotient below is defined.

Question. For every prime $p$ and every pair $(G,H)$ satisfying these assumptions, must \[ G/C_G(H^G) \] be a $p$-group?

Short English statement

Suppose a finitely generated group $G$ has a $p$-subgroup $H$ that contains no nontrivial normal subgroup of $G$ and satisfies $HX=XH$ for every subgroup $X\leq G$. Must the quotient of $G$ by the centralizer of the normal closure of $H$ be a $p$-group? The convention for infinite $p$-groups requires source review.

Sources

• Local UnsolvedMath registry, permanent ID $700826$, exact source code catalogue entry, title Kourovka Notebook Problem 12.6. The supplied record labels the problem open; this label is not an independently verified assessment.
• G. Busetto, Problem 12.6, The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), as attributed in the supplied record.
• The supplied background identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. No independent source consultation is asserted.