Kourovka Notebook Problem 17.128

Definitions and mathematical statement

Let $p$ be any prime, let $T$ be a finite group of order a power of $p$, and let $A\leq \operatorname{Aut}(T)$ satisfy $A\cong ({\mathbb{Z}}/p{\mathbb{Z}})^2$. Thus $A$ is an elementary abelian group of automorphisms of order $p^2$. Form the semidirect product $P=T\rtimes A$, with multiplication \[ (t,a)(u,b)=(t\,a(u),ab), \] and identify $T$ with $T\times\{1_A\}$. Suppose that \[ \operatorname{ord}_P(x)=p\qquad\text{for every }x\in P\setminus T. \] Must it follow that \[ t^p=1_T\qquad\text{for every }t\in T? \] This is the question whether $T$ has exponent $p$, where the exponent is the least positive integer $n$ such that $t^n=1_T$ for every $t\in T$. The subgroup condition on $A$ implies that $T$ is nontrivial, so the displayed conclusion is equivalent to exponent exactly $p$. No restriction to odd primes is supplied.

Short English statement

E. Jabara asks whether a finite $p$-group must have exponent $p$ if it admits an elementary abelian automorphism group of order $p^2$ such that every element outside the original group in the resulting semidirect product has order $p$.

Sources

• Local UnsolvedMath registry, supplied source record: permanent ID $701429$, exact source problem code catalogue entry, title Kourovka Notebook Problem 17.128. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), Problem 17.128; proposed by E. Jabara. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels the problem open. This is a supplied status, not an independently verified literature assessment; the cited editions have not been independently consulted for this draft.