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Published online by Cambridge University Press: 29 April 2018
Assume that $G$ is a finite group and $H$ is a 2-nilpotent Sylow tower Hall subgroup of $G$ such that if $x$ and $y$ are $G$-conjugate elements of $H\cap G^{\prime }$ of prime order or order 4, then $x$ and $y$ are $H$-conjugate. We prove that there exists a normal subgroup $N$ of $G$ such that $G=HN$ and $H\cap N=1$.
The first author is supported by the National Natural Science Foundation of China (no. 11401597). The second and third 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 second author is also supported by Prometeo/2017/057 of Generalitat (Valencian Community, Spain) and the third author is also supported by the China Scholarship Council, grant no. 201606890006.