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 |