Real elements and p-nilpotence of finite groups

Our first main result proves that every element of order 4 of a Sylow 2-subgroup S of a minimal non-2-nilpotent group G, is a real element of S. This allows to give a character-free proof of a theorem due to Isaacs and Navarro (see [9, Theorem B]). As an application, the authors show a common extension of the p-nilpotence criteria proved in [3] and [9]. The first and the second authors have been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economía y Competitividad, Spain, and FEDER, European Union. The first author has been also supported by a project from the National Natural Science Foundation of China (NSFC, No. 11271085) and a project of Natural Science Foundation…