Vipul: Created page with '{{wikilocal}} {{subgroup property}} ==Definition== A subgroup $H$ of a group $G$ is termed a '''finitarily hypernormalized subgroup''' if the iter...'

2009-03-27T13:06:52Z

Created page with '{{wikilocal}} {{subgroup property}} ==Definition== A subgroup <math>H</math> of a group <math>G</math> is termed a '''finitarily hypernormalized subgroup''' if the iter...'

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{{wikilocal}}
{{subgroup property}}

==Definition==

A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a '''finitarily hypernormalized subgroup''' if the iterated sequence of [[normalizer]]s, starting from <math>H</math>, reaches <math>G</math> in finitely many steps. In other words, define a sequence <math>H_i</math> by:

* <math>H_0 = H</math>.
* <math>H_{i+1} = N_G(H_i)</math>.

Then, <math>H</math> is <math>k</math>-hypernormalized if <math>H_k = G</math> for some natural number <math>k</math>. If there exists a natural number <math>k</math> such that <math>H</math> is <math>k</math>-hypernormalized, we say that <math>H</math> is finitarily hypernormalized.

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Normal subgroup]]
* [[Weaker than::2-hypernormalized subgroup]]
* [[Weaker than::Intermediately finitarily hypernormalized subgroup]]

===Weaker properties===

* [[Stronger than::Subnormal subgroup]]: In fact, a <math>k</math>-hypernormalized subgroup is <math>k</math>-subnormal. However, a <math>k</math>-subnormal subgroup need not be finitarily hypernormalized, and even if it is, it need not be <math>k</math>-normalized. {{proofat|[[Abnormal normalizer and 2-subnormal not implies normal]], [[normalizer of 2-subnormal subgroup may have arbitrarily large subnormal depth]]}}
* [[Stronger than::Hypernormalized subgroup]]

===Related group properties===

* [[Hypernormalizing group]] is a group in which every [[ascendant subgroup]] is [[hypernormalized subgroup|hypernormalized]]. A finite hypernormalizing group, or more generally, a [[slender group|slender]] hypernormalizing group, must have every [[subnormal subgroup]] is finitarily hypernormalized.
* In a [[nilpotent group]] every subgroup is hypernormalized, but a <math>k</math>-subnormal subgroup need not be <math>k</math>-hypernormalized.

==Metaproperties==

{{intransitive}}

{{not intsubcondn}}

Finitarily hypernormalized subgroup - Revision history

Vipul: Created page with '{{wikilocal}} {{subgroup property}} ==Definition== A subgroup H of a group G is termed a '''finitarily hypernormalized subgroup''' if the iter...'

Vipul: Created page with '{{wikilocal}} {{subgroup property}} ==Definition== A subgroup $H$ of a group $G$ is termed a '''finitarily hypernormalized subgroup''' if the iter...'