The nilpotency of some groups with all subgroups subnormal
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Let $G$ be a group with all subgroups subnormal. A normal subgroup $N$ of $G$ is said to be $G$-minimax if it has a finite $G$-invariant series whose factors are abelian and satisfy either $\max$-$G$ or $\min$-$G$. It is proved that if the normal closure of every element of $G$ is $G$-minimax then $G$ is nilpotent and the normal closure of every element is minimax. Further results of this type are also obtained.
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Kurdachenko, L. A.; Smith, H. «The nilpotency of some groups with all subgroups subnormal». Publicacions Matemàtiques, 1998, vol.VOL 42, núm. 2, p. 411-2, http://raco.cat/index.php/PublicacionsMatematiques/article/view/37943.
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