# 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 for all $H. 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.

Title normalizer condition NormalizerCondition 2013-03-22 16:14:41 2013-03-22 16:14:41 yark (2760) yark (2760) 6 yark (2760) Definition msc 20F19 normaliser condition LocallyNilpotentGroup N-group