Weakly $ \mathscr{H} $-embedded subgroups and nilpotency of fusion systems

Fawaz Aseeri; Julian Kaspczyk; Fawaz Aseeri; Julian Kaspczyk

doi:10.3934/math.2026995

AIMS Mathematics

2026, Volume 11, Issue 8: 24717-24732. doi: 10.3934/math.2026995

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Weakly $ \mathscr{H} $-embedded subgroups and nilpotency of fusion systems

Fawaz Aseeri ^{1
,
,},
Julian Kaspczyk ²

1.
Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia
2.
Berlin, Germany

Received: 03 June 2026 Revised: 20 July 2026 Accepted: 04 August 2026 Published: 12 August 2026
MSC : 20D10, 20D20

Let $ p $ be a prime and $ \mathcal{F} $ be a fusion system on a finite $ p $-group $ S $. We say that a subgroup $ Q $ of $ S $ is weakly $ \mathscr{H} $-embedded in $ \mathcal{F} $ if there is a strongly $ \mathcal{F} $-closed subgroup $ P $ of $ S $ such that both $ QP $ and $ Q \cap P $ are strongly $ \mathcal{F} $-closed. Using this concept, we prove some new characterizations of nilpotent fusion systems. Our work generalizes and extends some previously known results on weakly $ \mathscr{H} $-embedded subgroups of finite groups.
- finite group,
- fusion system,
- nilpotent,
- $ p $-nilpotent,
- strongly closed subgroup,
- $ \mathscr{H} $-subgroup,
- weakly $ \mathscr{H} $-embedded subgroup
Citation: Fawaz Aseeri, Julian Kaspczyk. Weakly $ \mathscr{H} $-embedded subgroups and nilpotency of fusion systems[J]. AIMS Mathematics, 2026, 11(8): 24717-24732. doi: 10.3934/math.2026995

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Abstract

Let $ p $ be a prime and $ \mathcal{F} $ be a fusion system on a finite $ p $-group $ S $. We say that a subgroup $ Q $ of $ S $ is weakly $ \mathscr{H} $-embedded in $ \mathcal{F} $ if there is a strongly $ \mathcal{F} $-closed subgroup $ P $ of $ S $ such that both $ QP $ and $ Q \cap P $ are strongly $ \mathcal{F} $-closed. Using this concept, we prove some new characterizations of nilpotent fusion systems. Our work generalizes and extends some previously known results on weakly $ \mathscr{H} $-embedded subgroups of finite groups.

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