Kourovka Notebook Problem 18.56

Definitions and mathematical statement

A finite $2$-group is a group $G$ with $|G|=2^n$ for some integer $n\geq 0$. For a subgroup $H\leq G$, define its core in $G$ by \[ H_G=\bigcap_{g\in G}gHg^{-1}. \] This is the largest normal subgroup of $G$ contained in $H$, and $|H/H_G|=[H:H_G]$.

The question proposed by G. Cutolo is whether, for every finite $2$-group $G$ with $|G|>2$, the condition \[ \forall H\leq G,\qquad [H:H_G]\leq 2 \] implies that there exists a subgroup $A\leq G$ such that \[ [G:A]=4 \qquad\text{and}\qquad \forall a,b\in A,\quad ab=ba. \] The index required is exactly $4$; the trivial subgroup is permitted. The supplied registry labels this question open; that label is not an independently verified assessment.

Short English statement

Must every finite $2$-group of order greater than $2$, in which each subgroup has index at most $2$ over its core, contain an abelian subgroup of index exactly $4$?

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701485$, exact source code catalogue entry, title "Kourovka Notebook Problem 18.56." The supplied status is open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 18.56, attributed to G. Cutolo, proposed in 2014. The supplied background identifies the 18th issue (2014) and the 21st issue edition (2026), with arXiv identifier 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.