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-group if and only if it is nilpotent.

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