Normal subgroup of nilpotent group - Revision history

Vipul: /* Weaker properties */

2013-02-24T06:24:21Z

Weaker properties

← Older revision		Revision as of 06:24, 24 February 2013
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	\| [[Stronger than::nilpotent-quotient subgroup]] \|\| normal subgroup such that the [[quotient group]] is a [[nilpotent group]]. \|\| \|\| \|\| {{intermediate notions short\|nilpotent-quotient subgroup\|normal subgroup of nilpotent group}}		\| [[Stronger than::nilpotent-quotient subgroup]] \|\| normal subgroup such that the [[quotient group]] is a [[nilpotent group]]. \|\| \|\| \|\| {{intermediate notions short\|nilpotent-quotient subgroup\|normal subgroup of nilpotent group}}
	\|-		\|-
	\| [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] \|\| if the whole group and the normal subgroup are powered over a prime, so is the quotient group. \|\| two proofs:<br>[[normal subgroup contained in the hypercenter satisfies a subgroup-to-quotient powering-invariance implication\|via normal subgroup contained in the hypercenter]]<br>[[nilpotent-quotient implies subgroup-to-quotient powering-invariance implication\|via nilpotent-quotient]] \|\| \|\| {{intermediate notions short\|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication\|normal subgroup of nilpotent group}}		\| [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] \|\| if the whole group and the normal subgroup are powered over a prime, so is the quotient group. \|\| two proofs:<br>[[normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication\|via normal subgroup contained in the hypercenter]]<br>[[nilpotent-quotient implies subgroup-to-quotient powering-invariance implication\|via nilpotent-quotient]] \|\| \|\| {{intermediate notions short\|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication\|normal subgroup of nilpotent group}}
	\|-		\|-
	\| [[Stronger than::subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|subgroup of nilpotent group\|normal subgroup of nilpotent group}}		\| [[Stronger than::subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|subgroup of nilpotent group\|normal subgroup of nilpotent group}}
	\|}		\|}

Vipul: /* Weaker properties */

2013-02-24T06:23:34Z

Weaker properties

← Older revision		Revision as of 06:23, 24 February 2013
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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::normal subgroup]] \|\| \|\| \|\| \|\|		\| [[Stronger than::normal subgroup]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subgroup\|normal subgroup of nilpotent group}}
	\|-		\|-
	\| [[Stronger than::normal subgroup contained in hypercenter]] \|\| \|\| \|\| \|\|		\| [[Stronger than::normal subgroup contained in hypercenter]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subgroup contained in the hypercenter\|normal subgroup of nilpotent group}}
	\|-		\|-
	\| [[Stronger than::subgroup of nilpotent group]] \|\| \|\| \|\| \|\|		\| [[Stronger than::nilpotent-quotient subgroup]] \|\| normal subgroup such that the [[quotient group]] is a [[nilpotent group]]. \|\| \|\| \|\| {{intermediate notions short\|nilpotent-quotient subgroup\|normal subgroup of nilpotent group}}
			\|-
			\| [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] \|\| if the whole group and the normal subgroup are powered over a prime, so is the quotient group. \|\| two proofs:<br>[[normal subgroup contained in the hypercenter satisfies a subgroup-to-quotient powering-invariance implication\|via normal subgroup contained in the hypercenter]]<br>[[nilpotent-quotient implies subgroup-to-quotient powering-invariance implication\|via nilpotent-quotient]] \|\| \|\| {{intermediate notions short\|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication\|normal subgroup of nilpotent group}}
			\|-
			\| [[Stronger than::subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|subgroup of nilpotent group\|normal subgroup of nilpotent group}}
	\|}		\|}

Vipul: /* Weaker properties */

2012-06-23T23:54:15Z

Weaker properties

← Older revision		Revision as of 23:54, 23 June 2012
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	\|-		\|-
	\| [[Stronger than::subgroup of nilpotent group]] \|\| \|\| \|\| \|\|		\| [[Stronger than::subgroup of nilpotent group]] \|\| \|\| \|\| \|\|
	\|-
	~~\| [[Stronger than::unipotent automorphism-invariant subgroup]] \|\| \|\| \|\| \|\|~~
	\|}		\|}

Vipul: /* Weaker properties */

2012-06-23T23:53:05Z

Weaker properties

← Older revision		Revision as of 23:53, 23 June 2012
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	\|-		\|-
	\| [[Stronger than::normal subgroup]] \|\| \|\| \|\| \|\|		\| [[Stronger than::normal subgroup]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Stronger than::normal subgroup contained in hypercenter]] \|\| \|\| \|\| \|\|
	\|-		\|-
	\| [[Stronger than::subgroup of nilpotent group]] \|\| \|\| \|\| \|\|		\| [[Stronger than::subgroup of nilpotent group]] \|\| \|\| \|\| \|\|

Vipul at 23:52, 23 June 2012

2012-06-23T23:52:10Z

← Older revision		Revision as of 23:52, 23 June 2012
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	The term '''normal subgroup of nilpotent group''' is used for a [[subgroup]] of a [[group]] where the whole group is a [[nilpotent group]] and the subgroup is a [[normal subgroup]].		The term '''normal subgroup of nilpotent group''' is used for a [[subgroup]] of a [[group]] where the whole group is a [[nilpotent group]] and the subgroup is a [[normal subgroup]].

			==Relation with other properties==

			===Stronger properties===

			{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::subgroup of abelian group]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Weaker than::normal subgroup of group of prime power order]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Weaker than::characteristic subgroup of nilpotent group]] \|\| \|\| \|\| \|\|
			\|}

			===Weaker properties===

			{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::normal subgroup]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Stronger than::subgroup of nilpotent group]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Stronger than::unipotent automorphism-invariant subgroup]] \|\| \|\| \|\| \|\|
			\|}

Vipul: Created page with "{{in-group subgroup property|normal subgroup|nilpotent group}} ==Definition== The term '''normal subgroup of nilpotent group''' is used for a subgroup of a group whe..."

2012-06-23T23:50:29Z

Created page with "{{in-group subgroup property|normal subgroup|nilpotent group}} ==Definition== The term '''normal subgroup of nilpotent group''' is used for a subgroup of a group whe..."

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{{in-group subgroup property|normal subgroup|nilpotent group}}

==Definition==

The term '''normal subgroup of nilpotent group''' is used for a [[subgroup]] of a [[group]] where the whole group is a [[nilpotent group]] and the subgroup is a [[normal subgroup]].