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A. Ballester-Bolinches, S. Camp-Mora, and F. Spagnuolo

On p-nilpotency of hyperfinite groups

Monatsh. Math., 176(4) (2015), 497–502

http://dx.doi.org/10.1007/s00605-014-0633-3

Abstract

Let p be a prime. We say that class X of hyperfinite p-groups determines p-nilpotency locally if every finite group G with a Sylow p-subgroup P in X is p-nilpotent if and only if 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 X, then G is p-nilpotent if and only if N_G(P) is p-nilpotent provided that X is closed under taking subgroups and epimorphic images. If 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.

2010 Mathematics subject classification: 20E15, 20F19, 20F22

Keywords: locally finite group; hyperfinite group; p-nilpotency