Fusion systems on a Sylow \(p\)-subgroup of \(SU_4(p)\). (English) Zbl 1490.20019
It is in general not easy to determine fusion systems on large \(p\)-groups. The paper finds all saturated fusion systems on a Sylow \(p\)-subgroup \(S\) of \(\mathrm{SU}_4(p)\) for odd \(p\), where \(S\) has order \(p^6\). Recall that a saturated fusion system is “realizable” if it is isomorphic to \(\mathcal{F}_S(G)\) for some finite group \(G\) containing \(S\) as a Sylow subgroup; otherwise, it is called “exotic”. We can see many special properties of exotic fusion systems from different perspectives. Thus, for a \(p\)-group \(S\), it is a focus of interest to determine whether there are exotic fusion systems on \(S\). The paper also proves that none of the fusion systems on a Sylow \(p\)-subgroup of \(\mathrm{SU}_4(p)\) is exotic, where \(p\) is an odd prime. Owing to previous work by B. Oliver [Mem. Am. Math. Soc. 1131, v, 100 p. (2016; Zbl 1382.20023)] and the work by E. Baccanelli et al. [J. Group Theory 22, No. 4, 689–711 (2019; Zbl 1468.20039)], the paper mainly focuses on the case when \(p\ge 5\).
Reviewer: Zhencai Shen (Beijing)
MSC:
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D05 | Finite simple groups and their classification |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
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