Vipul: /* Relation with other properties */

2008-09-03T17:28:02Z

Relation with other properties

← Older revision		Revision as of 17:28, 3 September 2008
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	===Stronger properties===		===Stronger properties===

	* [[Finitarily hypernormalized subgroup]]		* [[Weaker than::Finitarily hypernormalized subgroup]]
	* [[Normal subgroup]]		* [[Weaker than::Normal subgroup]]
			* [[Weaker than::2-hypernormalized subgroup]]

	===Weaker properties===		===Weaker properties===

	* [[Ascendant subgroup]]		* [[Stronger than::Ascendant subgroup]]
	* [[Serial subgroup]]		* [[Stronger than::Serial subgroup]]

	===Related group properties===		===Related group properties===

Vipul: 1 revision

2008-05-07T23:43:18Z

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Vipul at 01:17, 28 February 2007

2007-02-28T01:17:06Z

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{{subgroup property}}

{{variationof|normality}}

==Definition==

===Symbol-free definition===

A [[subgroup]] of a [[group]] is said to be '''hypernormalized''' if its [[hypernormalizer]] is the whole group. Here, the hypernormalizer is the limit of the transfinite ascending chain that begins with the subgroup and where each successor is the normalizer of its predecessor in the whole group.

If the hypernormalizer sequence reachse the whole group in finitely many steps, we call the subgroup '''finitarily hypernormalized'''.

===Definition with symbols===

A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is said to be '''hypernormalized''' if the sequence <math>H_{\alpha}</math> ends at <math>G</math>, where <math.H_\alpha</math> is defined as follows:

* <math>H_0 = H</math>
* <math>H_{\alpha + 1} = N_G(H_{\alpha})</math>
* <math>H_\alpha</math> is the ascending chain union of <math>H_\beta</math> for <matH>\beta < \alpha</math> when <math>\alpha</math> is a limit ordinal

If <math>H_r = G</math> for a finite integer <math>r</math>, we say that <math>H</math> is '''finiarily hypernormalized''' in <math>G</math>.

==Relation with other properties==

===Stronger properties===

* [[Finitarily hypernormalized subgroup]]
* [[Normal subgroup]]

===Weaker properties===

* [[Ascendant subgroup]]
* [[Serial subgroup]]

===Related group properties===

* [[HN-group]] is a group where every subgroup is hypernormalized.

==Metaproperties==

{{intransitive}}

A hypernormalized subgroup of a hypernormalized subgroup need not be hypernormalized. This can be seen from the fact that there are [[subnormal subgroup]]s which are not hypernormalized.

Hypernormalized subgroup - Revision history

Vipul: /* Relation with other properties */

Vipul: 1 revision

Vipul at 01:17, 28 February 2007