Central factor of a loop - Revision history

Vipul: /* Weaker properties */

2011-08-21T05:50:08Z

Weaker properties

← Older revision		Revision as of 05:50, 21 August 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! 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::Lagrange-like subloop]] \|\| \|\| \|\| \|\|
	\|-		\|-
	\| [[Stronger than::right-transitively normal subloop]] \|\| \|\| \|\| \|\|		\| [[Stronger than::right-transitively normal subloop]] \|\| \|\| \|\| \|\|

Vipul: /* Relation with other properties */

2011-08-21T05:43:07Z

Relation with other properties

← Older revision		Revision as of 05:43, 21 August 2011
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	\| [[Stronger than::normal subloop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subloop\|central factor of a loop}}		\| [[Stronger than::normal subloop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subloop\|central factor of a loop}}
	\|}		\|}

			===Incomparable properties===

			* [[Lagrange-like subloop]]: See [[central factor not implies Lagrange-like]]

Vipul: /* Weaker properties */

2011-08-21T05:40:30Z

Weaker properties

← Older revision		Revision as of 05:40, 21 August 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! 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::Lagrange-like subloop]] \|\| \|\| \|\| \|\|~~
	\|-		\|-
	\| [[Stronger than::right-transitively normal subloop]] \|\| \|\| \|\| \|\|		\| [[Stronger than::right-transitively normal subloop]] \|\| \|\| \|\| \|\|

Vipul: /* Weaker properties */

2011-08-20T06:41:41Z

Weaker properties

← Older revision		Revision as of 06:41, 20 August 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! 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::Lagrange-like subloop]] \|\| \|\| \|\| \|\|
	\|-		\|-
	\| [[Stronger than::right-transitively normal subloop]] \|\| \|\| \|\| \|\|		\| [[Stronger than::right-transitively normal subloop]] \|\| \|\| \|\| \|\|

Vipul at 06:33, 20 August 2011

2011-08-20T06:33:02Z

← Older revision		Revision as of 06:33, 20 August 2011
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	old specific context = subgroup\|		old specific context = subgroup\|
	new specific context = subloop\|		new specific context = subloop\|
	property = central factor}}		old property = central factor}}

	==Definition==		==Definition==

Vipul at 06:31, 20 August 2011

2011-08-20T06:31:23Z

← Older revision		Revision as of 06:31, 20 August 2011
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	{{subloop property}}		{{subloop property}}
			{{analogue of property\|
			old generic context = group\|
			new generic context = loop\|
			old specific context = subgroup\|
			new specific context = subloop\|
			property = central factor}}

	==Definition==		==Definition==

Vipul: /* Weaker properties */

2011-08-20T06:28:52Z

Weaker properties

← Older revision		Revision as of 06:28, 20 August 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! 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::right-transitively normal subloop]] \|\| \|\| \|\| \|\|
	\|-		\|-
	\| [[Stronger than::normal subloop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subloop\|central factor of a loop}}		\| [[Stronger than::normal subloop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subloop\|central factor of a loop}}
	\|}		\|}

Vipul: /* Relation with other properties */

2011-08-20T06:27:11Z

Relation with other properties

← Older revision		Revision as of 06:27, 20 August 2011
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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
	\|-		\|-
	\| [[Weaker than::direct factor of a loop]] \|\| \|\| \|\| \|\|		\| [[Weaker than::direct factor of a loop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|central factor of a loop\|direct factor of a loop}}
	\|-		\|-
	\| [[Weaker than::cocentral subloop]] \|\| \|\| \|\| \|\|		\| [[Weaker than::cocentral subloop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|central factor of a loop\|cocentral subloop}}
	\|-		\|-
	\| [[Weaker than::central subloop]] \|\| \|\| \|\| \|\|		\| [[Weaker than::central subloop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|central factor of a loop\|central subloop}}
	\|}		\|}

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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 subloop]] \|\| \|\| \|\| \|\|		\| [[Stronger than::normal subloop]] \|\| \|\| \|\| \|\| {{intermediate notions short\|normal subloop\|central factor of a loop}}
	\|}		\|}

Vipul: Created page with "{{subloop property}} ==Definition== A subloop $S$ of a loop $L$ is termed a '''central factor''' if we can write: $L = SC_L(S)$ where..."

2011-08-20T06:24:27Z

Created page with "{{subloop property}} ==Definition== A subloop <math>S</math> of a loop <math>L</math> is termed a '''central factor''' if we can write: <math>L = SC_L(S)</math> where..."

New page

{{subloop property}}

==Definition==

A [[subloop]] <math>S</math> of a [[loop]] <math>L</math> is termed a '''central factor''' if we can write:

<math>L = SC_L(S)</math>

where <math>C_L(S)</math> is the [[centralizer of a loop|centralizer]] of <math>S</math> in <math>L</math>, i.e., the elements that both commute and associate with <math>S</math>.

==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::direct factor of a loop]] || || || ||
|-
| [[Weaker than::cocentral subloop]] || || || ||
|-
| [[Weaker than::central subloop]] || || || ||
|}

===Weaker properties===

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

Central factor of a loop - Revision history

Vipul: /* Weaker properties */

Vipul: /* Relation with other properties */

Vipul: /* Weaker properties */

Vipul: /* Weaker properties */

Vipul at 06:33, 20 August 2011

Vipul at 06:31, 20 August 2011

Vipul: /* Weaker properties */

Vipul: /* Relation with other properties */

Vipul: Created page with "{{subloop property}} ==Definition== A subloop S of a loop L is termed a '''central factor''' if we can write: L = SC_L(S) where..."

Vipul: Created page with "{{subloop property}} ==Definition== A subloop $S$ of a loop $L$ is termed a '''central factor''' if we can write: $L = SC_L(S)$ where..."