characterization of finite nilpotent groups
Let $G$ be a finite group^{}. The following are equivalent^{}:

1.
$G$ is nilpotent^{}.
 2.

3.
Every proper subgroup^{} of $G$ is properly contained in its normalizer^{}.

4.
Every maximal subgroup of $G$ is normal.

5.
Every Sylow subgroup of $G$ is normal.

6.
$G$ is a direct product^{} (http://planetmath.org/DirectProductAndRestrictedDirectProductOfGroups) of $p$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  20130322 13:16:24 
Last modified on  20130322 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 