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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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new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

## 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 ✓