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arXiv · 2609.22764

On Intersections of System Normalizers

Ting Gong·Yuan Lu·Yong Yang·Michael Ruofan Zeng
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Abstract

We give a negative answer to Problem 17.39 of the Kourovka Notebook. For every odd prime power $q$ and every $m\geq1$, we construct a finite metabelian group $G_{m,q}$ with $Z_\infty(G_{m,q})=1$ and $|Φ(G_{m,q})|=q$ for which the least number of system normalizers with trivial intersection is $m+1$. Thus the number is unbounded even when the order of the Frattini subgroup is fixed.

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Ting Gong, Yuan Lu, Yong Yang, Michael Ruofan Zeng. 2026-09-19. On Intersections of System Normalizers. https://arxiv.org/abs/2609.22764

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