characterization of finite nilpotent groups

Let G be a finite groupMathworldPlanetmath. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    G is nilpotentPlanetmathPlanetmathPlanetmath.

  2. 2.

    Every subgroupMathworldPlanetmathPlanetmath ( of G is subnormal.

  3. 3.

    Every proper subgroupMathworldPlanetmath of G is properly contained in its normalizerMathworldPlanetmath.

  4. 4.

    Every maximal subgroup of G is normal.

  5. 5.

    Every Sylow subgroup of G is normal.

  6. 6.

    G is a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ( of p-groups (

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