Normal core of normalizer: Difference between revisions

Latest revision as of 01:02, 12 January 2010

This article defines a subgroup operator related to the subgroup property normal subgroup. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.

Definition

The normal core of normalizer of a subgroup $H$ of a group $G$ is defined in the following equivalent ways:

It is the largest normal subgroup of $G$ that normalizes $H$ . In other words, it is the largest normal subgroup of $G$ contained in the normalizer $N_{G} (H)$ .
It is the normal core of the normalizer $N_{G} (H)$ in $G$ .

$H$ is contained in the normal core of normalizer of $H$ if and only if $H$ is a 2-subnormal subgroup of $G$ . In this case, if $K$ is the normal core of normalizer of $H$ , then the ascending chain $H \leq K \leq G$ is the unique fastest ascending subnormal series for $H$ in $G$ . Further information: 2-subnormal subgroup has a unique fastest ascending subnormal series

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+{{subgroup operator related to|normal subgroup}}
 ==Definition==
 The '''normal core of normalizer''' of a [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is defined in the following equivalent ways:
-# It is the largest [[normal subgroup]] of <math>G</math> that normalizes <math>H</math. In other words, it is the largest [[normal subgroup]] of <math>G</math> contained in the [[defining ingredient::normalizer of a subgroup|normalizer]] <math>N_G(H)</math>.
+# It is the largest [[normal subgroup]] of <math>G</math> that normalizes <math>H</math>. In other words, it is the largest [[normal subgroup]] of <math>G</math> contained in the [[defining ingredient::normalizer of a subgroup|normalizer]] <math>N_G(H)</math>.
 # It is the [[defining ingredient::normal core]] of the [[normalizer]] <math>N_G(H)</math> in <math>G</math>.
 <math>H</math> is contained in the normal core of normalizer of <math>H</math> if and only if <math>H</math> is a [[2-subnormal subgroup]] of <math>G</math>. In this case, if <math>K</math> is the normal core of normalizer of <math>H</math>, then the ascending chain <math>H \le K \le G</math> is the unique fastest ascending subnormal series for <math>H</math> in <math>G</math>. {{further|[[2-subnormal subgroup has a unique fastest ascending subnormal series]]}}