Subnormalizer: Difference between revisions

Revision as of 16:39, 30 March 2009

Definition

Suppose $H$ is a subgroup of a group $G$ . A subnormalizer of $H$ in $G$ is a subgroup $K$ of $G$ containing $H$ such that $H$ is a subnormal subgroup of $K$ , and further, if $H\leq L\leq G$ is such that $H$ is subnormal in $L$ , then $L\leq K$ .

Since subnormality is not upper join-closed, not every subgroup need have a subnormalizer.

The term subnormalizer is also sometimes used for the subnormalizer subset, which is the largest subset in which the given subgroup is subnormal. When a subnormalizer exists in the sense described here, it coincides with the subnormalizer subset; however, the subnormalizer subset always exists, while the subnormalizer (subgroup) need not.

Revision as of 00:43, 22 February 2009 (view source) Vipul (talk \| contribs) (New page: ==Definition== Suppose <math>H</math> is a subgroup of a group <math>G</math>. A '''subnormalizer''' of <math>H</math> in <math>G</math> is a subgroup <math>K</math> of <math>G</m...)		Revision as of 16:39, 30 March 2009 (view source) Vipul (talk \| contribs) No edit summary Newer edit →
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	Since [[subnormality is not upper join-closed]], not every subgroup need have a subnormalizer.		Since [[subnormality is not upper join-closed]], not every subgroup need have a subnormalizer.

			The term '''subnormalizer''' is also sometimes used for the [[subnormalizer subset]], which is the largest ''subset'' in which the given subgroup is subnormal. When a subnormalizer exists in the sense described here, it coincides with the subnormalizer subset; however, the subnormalizer subset always exists, while the subnormalizer (subgroup) need not.