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 satisfies the normalizer condition if and only if for all . 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 |
---|---|
Canonical name | NormalizerCondition |
Date of creation | 2013-03-22 16:14:41 |
Last modified on | 2013-03-22 16:14:41 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 6 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 20F19 |
Synonym | normaliser condition |
Related topic | LocallyNilpotentGroup |
Defines | N-group |