Quotient-powering-invariant subgroup - Revision history

Vipul: /* Metaproperties */

2013-04-01T01:38:24Z

Metaproperties

← Older revision		Revision as of 01:38, 1 April 2013
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	\| [[satisfies metaproperty::quotient-transitive subgroup property]] \|\| Yes \|\| [[quotient-powering-invariance is quotient-transitive]] \|\| If <math>H \le K \le G</math> are such that <math>H</math> is quotient-powering-invariant in <math>G</math> and <math>K/H</matH> is quotient-powering-invariant in <math>G/H</math>, then <math>K</math> is quotient-powering-invariant in <math>G</math>.		\| [[satisfies metaproperty::quotient-transitive subgroup property]] \|\| Yes \|\| [[quotient-powering-invariance is quotient-transitive]] \|\| If <math>H \le K \le G</math> are such that <math>H</math> is quotient-powering-invariant in <math>G</math> and <math>K/H</matH> is quotient-powering-invariant in <math>G/H</math>, then <math>K</math> is quotient-powering-invariant in <math>G</math>.
	\|-		\|-
	\| [[satisfies ~~property~~::union-closed subgroup property]] \|\| Yes \|\| [[quotient-powering-invariance is union-closed]] \|\| If <math>H_i, i \in I</math> are all quotient-powering-invariant subgroups of a group <math>G</math>, and their set-theoretic union <math>\bigcup_{i \in I} H_i</math> is a subgroup <math>H</math>, then <math>H</math> is also a quotient-powering-invariant subgroup of <math>G</math>.		\| [[satisfies metaproperty::union-closed subgroup property]] \|\| Yes \|\| [[quotient-powering-invariance is union-closed]] \|\| If <math>H_i, i \in I</math> are all quotient-powering-invariant subgroups of a group <math>G</math>, and their set-theoretic union <math>\bigcup_{i \in I} H_i</math> is a subgroup <math>H</math>, then <math>H</math> is also a quotient-powering-invariant subgroup of <math>G</math>.
	\|}		\|}

Vipul: /* Metaproperties */

2013-04-01T01:38:05Z

Metaproperties

← Older revision		Revision as of 01:38, 1 April 2013
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	\|-		\|-
	\| [[satisfies metaproperty::quotient-transitive subgroup property]] \|\| Yes \|\| [[quotient-powering-invariance is quotient-transitive]] \|\| If <math>H \le K \le G</math> are such that <math>H</math> is quotient-powering-invariant in <math>G</math> and <math>K/H</matH> is quotient-powering-invariant in <math>G/H</math>, then <math>K</math> is quotient-powering-invariant in <math>G</math>.		\| [[satisfies metaproperty::quotient-transitive subgroup property]] \|\| Yes \|\| [[quotient-powering-invariance is quotient-transitive]] \|\| If <math>H \le K \le G</math> are such that <math>H</math> is quotient-powering-invariant in <math>G</math> and <math>K/H</matH> is quotient-powering-invariant in <math>G/H</math>, then <math>K</math> is quotient-powering-invariant in <math>G</math>.
			\|-
			\| [[satisfies property::union-closed subgroup property]] \|\| Yes \|\| [[quotient-powering-invariance is union-closed]] \|\| If <math>H_i, i \in I</math> are all quotient-powering-invariant subgroups of a group <math>G</math>, and their set-theoretic union <math>\bigcup_{i \in I} H_i</math> is a subgroup <math>H</math>, then <math>H</math> is also a quotient-powering-invariant subgroup of <math>G</math>.
	\|}		\|}

Vipul: /* Stronger properties */

2013-02-16T20:34:08Z

Stronger properties

← Older revision		Revision as of 20:34, 16 February 2013
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	\| [[Weaker than::finite normal subgroup]] \|\| the normal subgroup is finite \|\| [[finite normal implies quotient-powering-invariant]] \|\| \|\|		\| [[Weaker than::finite normal subgroup]] \|\| the normal subgroup is finite \|\| [[finite normal implies quotient-powering-invariant]] \|\| \|\|
	\|-		\|-
	\| [[Weaker than::~~direct factor~~]] \|\| normal subgroup ~~with normal complement~~ \|\| ~~(via complemented normal)~~ \|\| ~~(via complemented~~ normal) \|\| {{intermediate notions short~~\|direct factor~~\|quotient-powering-invariant subgroup}}		\| [[Weaker than::endomorphism kernel]] \|\| normal subgroup that is the kernel of an [[endomorphism]] \|\| [[endomorphism kernel implies quotient-powering-invariant]] \|\|any normal subgroup of a finite group that is not an endomorphism kernel works. \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|endomorphism kernel}}
	\|-		\|-
	\| [[Weaker than::complemented normal subgroup]] \|\| normal subgroup with a (possibly non-normal) complement \|\| [[complemented normal implies quotient-powering-invariant]] \|\| \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|complemented normal subgroup}}		\| [[Weaker than::complemented normal subgroup]] \|\| normal subgroup with a (possibly non-normal) complement \|\| (via endomorphism kernel, see also [[complemented normal implies quotient-powering-invariant\|direct proof]]) \|\| \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|complemented normal subgroup}}
	\|-		\|-
	\| [[Weaker than::~~endomorphism kernel~~]] \|\| normal subgroup ~~that is the kernel of an [[endomorphism]]~~ \|\| ~~[[endomorphism kernel implies quotient-powering-invariant]]~~ \|\|~~any~~ normal ~~subgroup of a finite group that is not an endomorphism kernel works.~~ \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|~~endomorphism kernel~~}}		\| [[Weaker than::direct factor]] \|\| normal subgroup with normal complement \|\| (via complemented normal) \|\| (via complemented normal) \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|direct factor}}
	\|-		\|-
	\| [[Weaker than::characteristic subgroup of abelian group]] \|\| [[characteristic subgroup]] and the whole group is an [[abelian group]] \|\| [[characteristic subgroup of abelian group is quotient-powering-invariant]] \|\| \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|characteristic subgroup of abelian group}}		\| [[Weaker than::characteristic subgroup of abelian group]] \|\| [[characteristic subgroup]] and the whole group is an [[abelian group]] \|\| [[characteristic subgroup of abelian group is quotient-powering-invariant]] \|\| \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|characteristic subgroup of abelian group}}

Vipul: /* Weaker properties */

2013-02-16T20:30:18Z

Weaker properties

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	\|-		\|-
	\| [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication\|quotient-powering-invariant subgroup}}		\| [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication\|quotient-powering-invariant subgroup}}
			\|-
			\| [[Stronger than::normal subgroup]] \|\| \|\| (by definition) \|\| (via powering-invariant normal) \|\| {{intermediate notions short\|normal subgroup\|quotient-powering-invariant subgroup}}
	\|}		\|}

Vipul: /* Weaker properties */

2013-02-16T20:24:03Z

Weaker properties

← Older revision		Revision as of 20:24, 16 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::powering-invariant subgroup]] \|\| \|\| [[quotient-powering-invariant implies powering-invariant]] \|\| [[powering-invariant not implies quotient-powering-invariant]] \|\| {{intermediate notions short\|powering-invariant subgroup\|quotient-powering-invariant subgroup}}		\| [[Stronger than::powering-invariant normal subgroup]] \|\| \|\| [[quotient-powering-invariant implies powering-invariant]] \|\| [[powering-invariant and normal not implies quotient-powering-invariant]] \|\| {{intermediate notions short\|powering-invariant normal subgroup\|quotient-powering-invariant subgroup}}
			\|-
			\| [[Stronger than::powering-invariant subgroup]] \|\| \|\| [[quotient-powering-invariant implies powering-invariant]] \|\| (via powering-invariant normal) \|\| {{intermediate notions short\|powering-invariant subgroup\|quotient-powering-invariant subgroup}}
	\|-		\|-
	\| [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication\|quotient-powering-invariant subgroup}}		\| [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication\|quotient-powering-invariant subgroup}}

Vipul: /* Stronger properties */

2013-02-16T18:06:21Z

Stronger properties

← Older revision		Revision as of 18:06, 16 February 2013
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	\|-		\|-
	\| [[Weaker than::complemented normal subgroup]] \|\| normal subgroup with a (possibly non-normal) complement \|\| [[complemented normal implies quotient-powering-invariant]] \|\| \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|complemented normal subgroup}}		\| [[Weaker than::complemented normal subgroup]] \|\| normal subgroup with a (possibly non-normal) complement \|\| [[complemented normal implies quotient-powering-invariant]] \|\| \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|complemented normal subgroup}}
			\|-
			\| [[Weaker than::endomorphism kernel]] \|\| normal subgroup that is the kernel of an [[endomorphism]] \|\| [[endomorphism kernel implies quotient-powering-invariant]] \|\|any normal subgroup of a finite group that is not an endomorphism kernel works. \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|endomorphism kernel}}
	\|-		\|-
	\| [[Weaker than::characteristic subgroup of abelian group]] \|\| [[characteristic subgroup]] and the whole group is an [[abelian group]] \|\| [[characteristic subgroup of abelian group is quotient-powering-invariant]] \|\| \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|characteristic subgroup of abelian group}}		\| [[Weaker than::characteristic subgroup of abelian group]] \|\| [[characteristic subgroup]] and the whole group is an [[abelian group]] \|\| [[characteristic subgroup of abelian group is quotient-powering-invariant]] \|\| \|\| {{intermediate notions short\|quotient-powering-invariant subgroup\|characteristic subgroup of abelian group}}

Vipul at 02:21, 13 February 2013

2013-02-13T02:21:03Z

← Older revision		Revision as of 02:21, 13 February 2013
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	{{subgroup property}}		{{subgroup property}}
	{{wikilocal}}		{{wikilocal}}
			{{subgroup property conjunction\|powering-invariant subgroup\|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication}}
	==Definition==		==Definition==

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	\| [[Stronger than::powering-invariant subgroup]] \|\| \|\| [[quotient-powering-invariant implies powering-invariant]] \|\| [[powering-invariant not implies quotient-powering-invariant]] \|\| {{intermediate notions short\|powering-invariant subgroup\|quotient-powering-invariant subgroup}}		\| [[Stronger than::powering-invariant subgroup]] \|\| \|\| [[quotient-powering-invariant implies powering-invariant]] \|\| [[powering-invariant not implies quotient-powering-invariant]] \|\| {{intermediate notions short\|powering-invariant subgroup\|quotient-powering-invariant subgroup}}
	\|-		\|-
	\| [[Stronger than::normal subgroup ~~with a~~ subgroup-to-quotient powering-invariance implication]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subgroup ~~with a~~ subgroup-to-quotient powering-invariance implication\|quotient-powering-invariant subgroup}}		\| [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication\|quotient-powering-invariant subgroup}}
	\|}		\|}

	===Properties whose conjunction with powering-invariance implies quotient-powering-invariance===		===Properties whose conjunction with powering-invariance implies quotient-powering-invariance===

	The relevant subgroup property is [[normal subgroup ~~with a~~ subgroup-to-quotient powering-invariance implication]]. This property is implied both by being a [[central subgroup]] and by being a [[normal subgroup contained in the hypercenter]].		The relevant subgroup property is [[normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]]. In fact, the conjunction of this with [[powering-invariant subgroup]] precisely gives [[quotient-powering-invariant subgroup]].

			This property is implied both by being a [[central subgroup]] and by being a [[normal subgroup contained in the hypercenter]].

Vipul: /* Weaker properties */

2013-02-13T01:58:03Z

Weaker properties

← Older revision		Revision as of 01:58, 13 February 2013
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	\|-		\|-
	\| [[Stronger than::powering-invariant subgroup]] \|\| \|\| [[quotient-powering-invariant implies powering-invariant]] \|\| [[powering-invariant not implies quotient-powering-invariant]] \|\| {{intermediate notions short\|powering-invariant subgroup\|quotient-powering-invariant subgroup}}		\| [[Stronger than::powering-invariant subgroup]] \|\| \|\| [[quotient-powering-invariant implies powering-invariant]] \|\| [[powering-invariant not implies quotient-powering-invariant]] \|\| {{intermediate notions short\|powering-invariant subgroup\|quotient-powering-invariant subgroup}}
			\|-
			\| [[Stronger than::normal subgroup with a subgroup-to-quotient powering-invariance implication]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subgroup with a subgroup-to-quotient powering-invariance implication\|quotient-powering-invariant subgroup}}
	\|}		\|}

Vipul: /* Properties whose conjunction with powering-invariance implies quotient-powering-invariance */

2013-02-13T01:56:06Z

Properties whose conjunction with powering-invariance implies quotient-powering-invariance

← Older revision		Revision as of 01:56, 13 February 2013
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	===Properties whose conjunction with powering-invariance implies quotient-powering-invariance===		===Properties whose conjunction with powering-invariance implies quotient-powering-invariance===

	~~{\| class="sortable" border="1"~~		The relevant subgroup property is [[normal subgroup with a subgroup-to-quotient powering-invariance implication]]. This property is implied both by being a [[central subgroup]] and by being a [[normal subgroup contained in the hypercenter]].
	~~! Property !! Proof of conjunction statement~~
	\|-
	\| [[~~central~~ subgroup~~]] \|\| [[powering~~-~~invariant and central implies~~ quotient-powering-~~invariant~~]]
	\|-
	\| [[~~normal~~ subgroup ~~contained in the hypercenter~~]] \|\| [[normal subgroup contained in the hypercenter ~~that is powering-invariant is quotient-powering-invariant~~]]
	\|}

Vipul: /* Incomparable properties */

2013-02-12T02:04:07Z

Incomparable properties

@@ Line 51: / Line 51: @@
 |-
 | [[normal subgroup contained in the hypercenter]] || [[normal subgroup contained in the hypercenter that is powering-invariant is quotient-powering-invariant]]
 |}