Characteristic subgroup of nilpotent group - Revision history

Vipul: /* Stronger properties */

2016-10-19T15:43:31Z

Stronger properties

← Older revision		Revision as of 15:43, 19 October 2016
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	\| [[Weaker than::verbal subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|verbal subgroup of nilpotent group}}		\| [[Weaker than::verbal subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|verbal subgroup of nilpotent group}}
	\|-		\|-
	\| [[Weaker than::powering-invariant characteristic subgroup of nilpotent group]] \|\| \|\| \|\| ~~see~~ [[characteristic not implies powering-invariant in nilpotent group]] \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|powering-invariant characteristic subgroup of nilpotent group}}		\| [[Weaker than::powering-invariant characteristic subgroup of nilpotent group]] \|\| \|\| \|\| [[characteristic not implies powering-invariant in nilpotent group]] \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|powering-invariant characteristic subgroup of nilpotent group}}
	\|}		\|}

Vipul: /* Stronger properties */

2016-10-19T15:43:02Z

Stronger properties

← Older revision		Revision as of 15:43, 19 October 2016
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	\| [[Weaker than::verbal subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|verbal subgroup of nilpotent group}}		\| [[Weaker than::verbal subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|verbal subgroup of nilpotent group}}
	\|-		\|-
	\| [[Weaker than::powering-invariant characteristic subgroup of nilpotent group]] \|\| \|\| \|\| see [[characteristic ~~subgroup of nilpotent group~~ not implies powering-invariant]] \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|powering-invariant characteristic subgroup of nilpotent group}}		\| [[Weaker than::powering-invariant characteristic subgroup of nilpotent group]] \|\| \|\| \|\| see [[characteristic not implies powering-invariant in nilpotent group]] \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|powering-invariant characteristic subgroup of nilpotent group}}
	\|}		\|}

Vipul: /* Stronger properties */

2016-10-19T15:09:30Z

Stronger properties

← Older revision		Revision as of 15:09, 19 October 2016
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	\|-		\|-
	\| [[Weaker than::verbal subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|verbal subgroup of nilpotent group}}		\| [[Weaker than::verbal subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|verbal subgroup of nilpotent group}}
			\|-
			\| [[Weaker than::powering-invariant characteristic subgroup of nilpotent group]] \|\| \|\| \|\| see [[characteristic subgroup of nilpotent group not implies powering-invariant]] \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|powering-invariant characteristic subgroup of nilpotent group}}
	\|}		\|}

Vipul: /* Stronger properties */

2013-07-15T03:18:41Z

Stronger properties

← Older revision		Revision as of 03:18, 15 July 2013
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	\|-		\|-
	\| [[Weaker than::fully invariant subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|fully invariant subgroup of nilpotent group}}		\| [[Weaker than::fully invariant subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|fully invariant subgroup of nilpotent group}}
			\|-
			\| [[Weaker than::verbal subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|verbal subgroup of nilpotent group}}
	\|}		\|}

Vipul: /* Relation with other properties */

2013-07-15T03:13:56Z

Relation with other properties

← Older revision		Revision as of 03:13, 15 July 2013
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	\|-		\|-
	\| [[Weaker than::characteristic subgroup of abelian group]] \|\| \|\| follows from [[abelian implies nilpotent]] \|\| follows from [[nilpotent not implies abelian]] \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|characteristic subgroup of abelian group}}		\| [[Weaker than::characteristic subgroup of abelian group]] \|\| \|\| follows from [[abelian implies nilpotent]] \|\| follows from [[nilpotent not implies abelian]] \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|characteristic subgroup of abelian group}}
			\|-
			\| [[Weaker than::characteristic direct factor of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|characteristic direct factor of nilpotent group}}
			\|-
			\| [[Weaker than::fully invariant subgroup of nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|characteristic subgroup of nilpotent group\|fully invariant subgroup of nilpotent group}}
	\|}		\|}

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

2013-02-20T04:13:05Z

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

New page

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

==Definition==

The term '''characteristic subgroup of nilpotent group''' is used for a situation where the whole group is a [[nilpotent group]] and the subgroup is a [[characteristic 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::characteristic subgroup of abelian group]] || || follows from [[abelian implies nilpotent]] || follows from [[nilpotent not implies abelian]] || {{intermediate notions short|characteristic subgroup of nilpotent group|characteristic subgroup of abelian group}}
|}

===Weaker properties===

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