characterization of finite nilpotent groups

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

1. 1.

$G$ is nilpotent.

2. 2.

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

3. 3.

Every proper subgroup of $G$ is properly contained in its normalizer.

4. 4.

Every maximal subgroup of $G$ is normal.

5. 5.

Every Sylow subgroup of $G$ is normal.

6. 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