Normalizer condition implies locally nilpotent: Difference between revisions

Latest revision as of 23:11, 16 April 2017

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group satisfying normalizer condition) must also satisfy the second group property (i.e., locally nilpotent group)
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Statement

Suppose $G$ is a group satisfying normalizer condition: for any proper subgroup $H$ of $G$ , the normalizer $N_{G} (H)$ is strictly bigger than $H$ . Then, $G$ is a locally nilpotent group: any finitely generated subgroup of $G$ is nilpotent.

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 stronger = group satisfying normalizer condition|
 weaker = locally nilpotent group}}
+== Statement ==
+Suppose <math>G</math> is a [[group satisfying normalizer condition]]: for any proper subgroup <math>H</math> of <math>G</math>, the [[normalizer]] <math>N_G(H)</math> is strictly bigger than <math>H</math>. Then, <math>G</math> is a [[locally nilpotent group]]: any finitely generated subgroup of <math>G</math> is nilpotent.