characterization of finite nilpotent groups
Let be a finite group. The following are equivalent
:
- 1.
is nilpotent
.
- 2.
Every subgroup
(http://planetmath.org/Subgroup) of is subnormal.
- 3.
Every proper subgroup
of is properly contained in its normalizer
.
- 4.
Every maximal subgroup of is normal.
- 5.
Every Sylow subgroup of is normal.
- 6.
is a direct product
(http://planetmath.org/DirectProductAndRestrictedDirectProductOfGroups) of -groups (http://planetmath.org/PGroup4).
For proofs, see the article on finite nilpotent groups.
Condition 3 above is the normalizer condition.
Title | characterization of finite nilpotent groups |
Canonical name | CharacterizationOfFiniteNilpotentGroups |
Date of creation | 2013-03-22 13:16:24 |
Last modified on | 2013-03-22 13:16:24 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 11 |
Author | yark (2760) |
Entry type | Theorem |
Classification | msc 20D15 |
Classification | msc 20F18 |
Related topic | FiniteNilpotentGroups |
Related topic | NilpotentGroup |
Related topic | NormalizerCondition |
Related topic | SubnormalSubgroup |
Related topic | LocallyNilpotentGroup |