normalizer condition

A group is said to satisfy the normalizer condition if every proper subgroup is properly contained in its own normalizer.That is, a group $G$ satisfies the normalizer condition if and only if $H<N_{G}(H)$ for all $H<G$ .A group that satisfies the normalizer condition is sometimes called an N-group.

Every nilpotent group is an N-group, and every N-group is locally nilpotent.In particular, a finitely generated group is an N-groupif and only if it is nilpotent.

A group satisfies the normalizer condition if and only if all its subgroups are ascendant.

单词	NormalizerCondition
释义	normalizer condition A group is said to satisfy the normalizer condition if every proper subgroup is properly contained in its own normalizer.That is, a group $G$ satisfies the normalizer condition if and only if $H<N_{G}(H)$ for all $H<G$ .A group that satisfies the normalizer condition is sometimes called an N-group. Every nilpotent group is an N-group, and every N-group is locally nilpotent.In particular, a finitely generated group is an N-groupif and only if it is nilpotent. A group satisfies the normalizer condition if and only if all its subgroups are ascendant.
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