characterization of finite nilpotent groups

Let $G$ be a finite group. The following are equivalent:

1. $G$ is nilpotent.

2. Every subgroup (http://planetmath.org/Subgroup) of $G$ is subnormal.

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	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