On $$p$$ p -nilpotency of hyperfinite groups

Autor: Francesca Spagnuolo, S. Camp-Mora, Adolfo Ballester-Bolinches
Rok vydání: 2014
Předmět:
Zdroj: Monatshefte für Mathematik. 176:497-502
ISSN: 1436-5081
0026-9255
DOI: 10.1007/s00605-014-0633-3
Popis: Let \(p\) be a prime. We say that class \(\fancyscript{X}\) of hyperfinite \(p\)-groups determines\(p\)-nilpotency locally if every finite group \(G\) with a Sylow \(p\)-subgroup \(P\) in \(\fancyscript{X}\) is \(p\)-nilpotent if and only if \({{\mathrm{N}}}_{G}(P)\) is \(p\)-nilpotent. The results of this paper improve a recent result of Kurdachenko and Otal and show that if a hyperfinite group \(G\) has a pronormal Sylow \(p\)-subgroup in \(\fancyscript{X}\), then \(G\) is \(p\)-nilpotent if and only if \({{\mathrm{N}}}_G(P)\) is \(p\)-nilpotent provided that \(\fancyscript{X}\) is closed under taking subgroups and epimorphic images. If \(\fancyscript{X}\) is not closed under taking epimorphic images, we have to impose local \(p\)-solubility to \(G\). In this case, the hypothesis of pronormality can be removed.
Databáze: OpenAIRE
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