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# 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 of $p$-groups.

For proofs, see the article on finite nilpotent groups.

Condition 3 above is the normalizer condition.

Related:

FiniteNilpotentGroups , NilpotentGroup, NormalizerCondition, SubnormalSubgroup, LocallyNilpotentGroup

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

20D15*no label found*20F18

*no label found*

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## Recent Activity

Jul 5

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb

Jun 6

new question: difference of a function and a finite sum by pfb

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb

Jun 6

new question: difference of a function and a finite sum by pfb

## Corrections

20D15 by yark ✓

Other name by mathwizard ✘

Condition 3 by JadeNB ✓

title by mps ✓

proof reference by Algeboy ✓

Other name by mathwizard ✘

Condition 3 by JadeNB ✓

title by mps ✓

proof reference by Algeboy ✓