Kourovka Notebook Problem 21.35

Definitions and mathematical statement

Use the commutator convention $[u,v]=u^{-1}v^{-1}uv$. A multilinear commutator word is obtained recursively from distinct variables by repeatedly taking commutators of words involving disjoint sets of variables. Thus every variable occurs once in its commutator expression; different bracket arrangements are allowed. For a word $w=w(x_1,\ldots,x_r)$ and a group $G$, a $w$-value is an element $w(g_1,\ldots,g_r)$ with $g_i\in G$, and the verbal subgroup $w(G)$ is generated by all such values.

Fix a finite group $G$, a prime $p$, and such a word $w$. For an element $g$, let $o(g)$ be its order, and call this a $p'$-order if $p\nmid o(g)$. Assume that for every pair of $w$-values $x,y$ in $G$, \[ p\nmid o(x),\quad p\mid o(y)\quad\Longrightarrow\quad p\mid o(xy). \] Must $w(G)$ be $p$-nilpotent? A finite group $K$ is $p$-nilpotent here if it has a normal subgroup $N$ of order prime to $p$ with $K/N$ a $p$-group; equivalently it has a normal Hall $p'$-subgroup.

The order hypothesis is imposed on individual word values, not on all elements of the generated subgroup. The conclusion nevertheless concerns the entire verbal subgroup. Both $x$ and $y$ are chosen from the same word-value set, and their product need not itself be a word value. Multilinearity of $w$ remains an essential hypothesis of the question.

The editors mark this entry with an unverified AI solution, which their preface expressly distinguishes from a confirmed solution. The mathematical question is therefore retained without endorsing that proposed argument.

Short English statement

Does the specified product-order condition on multilinear commutator values force their verbal subgroup to have a normal Hall $p'$-subgroup?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2544 (KOU-21.35), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.35. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.