Kourovka Notebook Problem 20.48

Definitions and mathematical statement

Let $p$ be a prime and let $P$ be a nontrivial finite $p$-group. For this draft, a nontrivial partition of $P$ means a family $\mathcal P$ of at least two nonidentity proper subgroups such that \[ P=\bigcup_{U\in\mathcal P}U, \qquad U\cap V=\{1\}\quad\text{for distinct }U,V\in\mathcal P. \] Define the Hughes subgroup by \[ H_p(P)=\left\langle g\in P\mid g^p\ne 1\right\rangle. \] An automorphism $\alpha\in\operatorname{Aut}(P)$ is fixed-point-free if \[ \{x\in P\mid \alpha(x)=x\}=\{1\}. \] The statement that $P$ has exponent $p$ means that every element satisfies $x^p=1$ and some element has order $p$.

The question asks whether, for every prime $p$ and every nontrivial finite $p$-group $P$ admitting a nontrivial partition, the following implication holds: \[ \left(\exists\text{ a prime }r\;\exists\alpha\in\operatorname{Aut}(P): \operatorname{ord}(\alpha)=r \ \text{and}\ \{x\in P\mid\alpha(x)=x\}=\{1\}\right) \Longrightarrow \operatorname{exp}(P)=p. \] No relation between $r$ and $p$ is imposed in the supplied statement.

The supplied record identifies the partition hypothesis with $H_p(P)\ne P$. Its alternate formulation therefore replaces the partition hypothesis above by this proper-Hughes-subgroup condition. The source does not specify its convention for a nontrivial partition. Under the convention used here, the cyclic group of order $p$ has proper Hughes subgroup but has no nontrivial partition. Thus the asserted equivalence requires a boundary qualification or a different partition convention; the source's intended convention remains unresolved.

Short English statement

Must a finite $p$-group with a nontrivial partition have exponent $p$ whenever it admits a fixed-point-free automorphism of prime order? The source also gives the hypothesis as $H_p(P)\ne P$; the precise equivalence of these hypotheses has an unresolved small-order convention.

Sources

• Local UnsolvedMath registry, permanent ID $701706$, exact source problem code catalogue entry, title Kourovka Notebook Problem 20.48. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Attributed in the supplied record to M. Lewis, proposed in 2022; original problem number 20.48 in The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), with the 21st issue edition (2026), arXiv:1401.0300v48, also identified as a source. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels this problem open and gives an update timestamp of 2026-10-16T14:00:00Z. These are supplied registry metadata, not an independently verified assessment of the literature. The referenced editions and arXiv version have not been independently consulted for this draft.