Elliptic subgroup - Revision history

Vipul: /* Stronger properties */

2020-03-09T04:18:21Z

Stronger properties

← Older revision		Revision as of 04:18, 9 March 2020
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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::normal subgroup \|\| \|\| \|\| \|\| {{intermediate notions short\|elliptic subgroup\|normal subgroup}}		\| [[Weaker than::normal subgroup]] \|\| \|\| \|\| \|\| {{intermediate notions short\|elliptic subgroup\|normal subgroup}}
	\|-		\|-
	\| [[Weaker than::permutable subgroup]] \|\| permutes with every subgroup \|\| \|\| \|\| {{intermediate notions short\|elliptic subgroup\|permutable subgroup}}		\| [[Weaker than::permutable subgroup]] \|\| permutes with every subgroup \|\| \|\| \|\| {{intermediate notions short\|elliptic subgroup\|permutable subgroup}}

Vipul: /* Stronger properties */

2020-03-09T04:17:55Z

Stronger properties

← Older revision		Revision as of 04:17, 9 March 2020
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	===Stronger properties===		===Stronger properties===

	* [[~~Permutable~~ subgroup]]		{\| class="sortable" border="1"
	* [[~~Subgroup~~ of finite index]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::normal subgroup \|\| \|\| \|\| \|\| {{intermediate notions short\|elliptic subgroup\|normal subgroup}}
			\|-
			\| [[Weaker than::permutable subgroup]] \|\| permutes with every subgroup \|\| \|\| \|\| {{intermediate notions short\|elliptic subgroup\|permutable subgroup}}
			\|-
			\| [[Weaker than::subgroup of finite index]] \|\| \|\| \|\| \|\| {{intermediate notions short\|elliptic subgroup\|subgroup of finite index}}
			\|}

Vipul: /* Definition with symbols */

2009-03-09T19:32:04Z

Definition with symbols

← Older revision		Revision as of 19:32, 9 March 2009
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	A subgroup <math>H</math> of a group <math>G</math> is termed '''elliptic''' if for any subgroup <math>K</math> of <math>G</math>, <math>(H,K)</math> form an [[elliptic pair of subgroups]]. In other words, there exists an <math>n</math> such that:		A subgroup <math>H</math> of a group <math>G</math> is termed '''elliptic''' if for any subgroup <math>K</math> of <math>G</math>, <math>(H,K)</math> form an [[elliptic pair of subgroups]]. In other words, there exists an <math>n</math> such that:

	<math><H,K> = (HK)^n := HKHKHK \ldots HK</math>		<math>\langle H,K \rangle = (HK)^n := HKHKHK \ldots HK</math>

	where each is written <math>n</math> times.		where each is written <math>n</math> times.

Vipul at 19:31, 9 March 2009

2009-03-09T19:31:36Z

← Older revision		Revision as of 19:31, 9 March 2009
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	{{subgroup property}}		{{subgroup property}}
			{{finitarily tautological subgroup property}}

	==Definition==		==Definition==

Vipul: 1 revision

2008-05-07T23:27:19Z

1 revision

← Older revision	Revision as of 23:27, 7 May 2008
(No difference)

Vipul at 22:51, 20 September 2007

2007-09-20T22:51:44Z

New page

{{subgroup property}}

==Definition==

===Symbol-free definition===

A [[subgroup]] of a [[group]] is termed '''elliptic''' if it forms an [[elliptic pair of subgroups]] with every subgroup of the group.

===Definition with symbols===

A subgroup <math>H</math> of a group <math>G</math> is termed '''elliptic''' if for any subgroup <math>K</math> of <math>G</math>, <math>(H,K)</math> form an [[elliptic pair of subgroups]]. In other words, there exists an <math>n</math> such that:

<math><H,K> = (HK)^n := HKHKHK \ldots HK</math>

where each is written <math>n</math> times.

==Relation with other properties==

===Stronger properties===

* [[Permutable subgroup]]
* [[Subgroup of finite index]]