PreprintCC BY 4.0arXiv
On Intersections of System Normalizers
Abstract
We give a negative answer to Problem 17.39 of the Kourovka Notebook. For every odd prime power q and every m≥1, we construct a finite metabelian group Gm,q with Z∞(Gm,q)=1 and |Φ(Gm,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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BibTeX
@misc{gong2026intersections,
title = {{On Intersections of System Normalizers}},
author = {Ting Gong and Yuan Lu and Yong Yang and Michael Ruofan Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22764},
eprint = {2609.22764},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22764},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Intersections of System Normalizers
AU - Ting Gong
AU - Yuan Lu
AU - Yong Yang
AU - Michael Ruofan Zeng
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22764
UR - https://arxiv.org/abs/2609.22764
AB - We give a negative answer to Problem 17.39 of the Kourovka Notebook. For every odd prime power q and every m≥1, we construct a finite metabelian group G_m,q with Z_∞(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.
ER -