Subgroup of finite index - Revision history

Vipul: /* Weaker properties */

2020-03-09T15:25:00Z

Weaker properties

← Older revision		Revision as of 15:25, 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
	\|-		\|-
	\| [Stronger than::subgroup of finite double coset index]] \|\| has finitely many double cosets in the whole group \|\| (obvious, since double cosets are unions of cosets) \|\| counterexamples can be found using Borel subgroups in general linear groups \|\| {{intermediate notions short\|subgroup of finite double coset index\|subgroup of finite index}}		\| [[Stronger than::subgroup of finite double coset index]] \|\| has finitely many double cosets in the whole group \|\| (obvious, since double cosets are unions of cosets) \|\| counterexamples can be found using Borel subgroups in general linear groups \|\| {{intermediate notions short\|subgroup of finite double coset index\|subgroup of finite index}}
	\|-		\|-
	\| [[Stronger than::subgroup contained in finitely many intermediate subgroups]] \|\| the number of intermediate subgroups between the subgroup and whole group is finite \|\| (via finite double coset index) \|\| (via finite double coset index) \|\| {{intermediate notions short\|subgroup contained in finitely many intermediate subgroups\|subgroup of finite index}}		\| [[Stronger than::subgroup contained in finitely many intermediate subgroups]] \|\| the number of intermediate subgroups between the subgroup and whole group is finite \|\| (via finite double coset index) \|\| (via finite double coset index) \|\| {{intermediate notions short\|subgroup contained in finitely many intermediate subgroups\|subgroup of finite index}}
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	\| [[Stronger than::conjugate-commensurable subgroup]] \|\| its intersection with any of its [[conjugate subgroups]] has finite index in both \|\| (via nearly normal) \|\| (via nearly normal) \|\| {{intermediate notions short\|conjugate-commensurable subgroup\|subgroup of finite index}}		\| [[Stronger than::conjugate-commensurable subgroup]] \|\| its intersection with any of its [[conjugate subgroups]] has finite index in both \|\| (via nearly normal) \|\| (via nearly normal) \|\| {{intermediate notions short\|conjugate-commensurable subgroup\|subgroup of finite index}}
	\|-		\|-
	\| [[Stronger than::~~automorp~~-commensurable subgroup]] \|\| its intersection with any of its [[automorphic subgroups]] has finite index in both \|\| \|\| \|\| {{intermediate notions short\|automorph-commensurable subgroup\|subgroup of finite index}}		\| [[Stronger than::automorph-commensurable subgroup]] \|\| its intersection with any of its [[automorphic subgroups]] has finite index in both \|\| \|\| \|\| {{intermediate notions short\|automorph-commensurable subgroup\|subgroup of finite index}}
	\|}		\|}

Vipul: /* Relation with other properties */

2020-03-09T15:24:49Z

Relation with other properties

← Older revision		Revision as of 15:24, 9 March 2020
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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::~~Subgroup~~ of index two]]		{\| class="sortable" border="1"
	* [[Weaker than::~~Subgroup~~ of prime index]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Weaker than::~~Normal~~ subgroup of finite index]]		\|-
			\| [[Weaker than::subgroup of index two]] \|\| the index is two; i.e., it has just two cosets \|\| (obvious) \|\| take the trivial subgroup in any finite group of order greater than two \|\| {{intermediate notions short\|subgroup of finite index\|subgroup of index two}}
			\|-
			\| [[Weaker than::subgroup of prime index]] \|\| the index is a prime number \|\| (obvious) \|\| take the trivial subgroup in any finite group of non-prime order \|\| {{intermediate notions short\|subgroup of finite index\|subgroup of prime index}}
			\|-
			\| [[Weaker than::normal subgroup of finite index]] \|\| normal subgroup whose quotient group is finite \|\| (obvious) \|\| take any non-normal subgroup of a finite group \|\| {{intermediate notions short\|subgroup of finite index\|normal subgroup of finite index}}
			\|-
			\| [[Weaker than::subgroup of finite group]] \|\| the whole group is finite \|\| (obvious) \|\| any infinite group as subgroup of itself; <math>n\mathbb{Z}</math> as subgroup of <math>\mathbb{Z}</math> \|\| {{intermediate notions short\|subgroup of finite index\|subgroup of finite group}}
			\|}

	===Weaker properties===		===Weaker properties===

	* [[Stronger than::~~Subgroup~~ of finite double coset index]]		{\| class="sortable" border="1"
	* [[Stronger than::~~Subgroup~~ contained in finitely many intermediate subgroups]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Stronger than::~~Almost~~ normal subgroup]]		\|-
	* [[Stronger than::~~Nearly~~ normal subgroup]]		\| [Stronger than::subgroup of finite double coset index]] \|\| has finitely many double cosets in the whole group \|\| (obvious, since double cosets are unions of cosets) \|\| counterexamples can be found using Borel subgroups in general linear groups \|\| {{intermediate notions short\|subgroup of finite double coset index\|subgroup of finite index}}
			\|-
			\| [[Stronger than::subgroup contained in finitely many intermediate subgroups]] \|\| the number of intermediate subgroups between the subgroup and whole group is finite \|\| (via finite double coset index) \|\| (via finite double coset index) \|\| {{intermediate notions short\|subgroup contained in finitely many intermediate subgroups\|subgroup of finite index}}
			\|-
			\| [[Stronger than::almost normal subgroup]] \|\| its normalizer in the whole group has finite index \|\| (obvious) \|\| trivial subgroup in an infinite group \|\| {{intermediate notions short\|almost normal subgroup\|subgroup of finite index}}
			\|-
			\| [[Stronger than::nearly normal subgroup]] \|\| it has finite index in its normal closure \|\| (obvious) \|\| trivial subgroup in an infinite group \|\| {{intermediate notions short\|nearly normal subgroup\|subgroup of finite index}}
			\|-
			\| [[Stronger than::conjugate-commensurable subgroup]] \|\| its intersection with any of its [[conjugate subgroups]] has finite index in both \|\| (via nearly normal) \|\| (via nearly normal) \|\| {{intermediate notions short\|conjugate-commensurable subgroup\|subgroup of finite index}}
			\|-
			\| [[Stronger than::automorp-commensurable subgroup]] \|\| its intersection with any of its [[automorphic subgroups]] has finite index in both \|\| \|\| \|\| {{intermediate notions short\|automorph-commensurable subgroup\|subgroup of finite index}}
			\|}

	===Conjunction with other properties===		===Conjunction with other properties===

	* [[~~Normal~~ subgroup of finite index]]		* [[normal subgroup of finite index]]

	==Metaproperties==		==Metaproperties==

Vipul: /* Equivalent definitions in tabular format */

2020-03-09T15:13:41Z

Equivalent definitions in tabular format

← Older revision		Revision as of 15:13, 9 March 2020
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	! No. !! Shorthand !! A subgroup of a group is termed a subgroup of finite index if ... !! A subgroup <math>H</math> of a group <math>G</math> is termed a subgroup of finite index if ...		! No. !! Shorthand !! A subgroup of a group is termed a subgroup of finite index if ... !! A subgroup <math>H</math> of a group <math>G</math> is termed a subgroup of finite index if ...
	\|-		\|-
	\| 1 \|\| finite index \|\| its [[defining ingredient::index of a subgroup;1\|index]] in the whole group is finite. \|\| the index <math>\|G:H\|</math>, defined as the size of the coset space <math>[G:H]</math>, is finite.		\| 1 \|\| finite index \|\| its [[defining ingredient::index of a subgroup\|index]] in the whole group is finite. \|\| the index <math>\|G:H\|</math>, defined as the size of the coset space <math>[G:H]</math>, is finite.
	\|-		\|-
	\| 2 \|\| contains normal subgroup of finite index \|\| it contains a [[normal subgroup of finite index]]. \|\| there is a normal subgroup <math>N</math> of <math>G</math> such that <math>N \le H \le G</math> and the quotient group <math>G/N</math> is finite.		\| 2 \|\| contains normal subgroup of finite index \|\| it contains a [[normal subgroup of finite index]]. \|\| there is a normal subgroup <math>N</math> of <math>G</math> such that <math>N \le H \le G</math> and the quotient group <math>G/N</math> is finite.

Vipul: /* Definition */

2020-03-09T15:13:29Z

Definition

← Older revision		Revision as of 15:13, 9 March 2020
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	==Definition==		==Definition==

	===~~Symbol-free definition~~===		===Equivalent definitions in tabular format===

	A [[subgroup]] of a [[group]] is ~~said to be '''~~of finite index~~'''~~ if its [[index of a subgroup\|index]] in the whole group is finite, ~~or equivalently~~, if it ~~has only finitely many~~ [[~~coset~~]]s.		{\| class="sortable" border="1"
			! No. !! Shorthand !! A subgroup of a group is termed a subgroup of finite index if ... !! A subgroup <math>H</math> of a group <math>G</math> is termed a subgroup of finite index if ...
	~~===Definition with symbols===~~		\|-
			\| 1 \|\| finite index \|\| its [[defining ingredient::index of a subgroup;1\|index]] in the whole group is finite. \|\| the index <math>\|G:H\|</math>, defined as the size of the coset space <math>[G:H]</math>, is finite.
	A [[~~subgroup~~]] <math>H</math> ~~of a group~~ <math>G</math> is ~~said to be '''~~of finite index~~''' if~~ <math>[G~~:H]~~</math> ~~is finite~~.		\|-
			\| 2 \|\| contains normal subgroup of finite index \|\| it contains a [[normal subgroup of finite index]]. \|\| there is a normal subgroup <math>N</math> of <math>G</math> such that <math>N \le H \le G</math> and the quotient group <math>G/N</math> is finite.
			\|-
			\| 3 \|\| normal core has finite index \|\| its [[normal core]] in the whole group has finite index in the whole group. \|\| the normal core of <math>H</math> in <math>G</math> is a normal subgroup of finite index in <math>G</math>.
			\|}

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Relation with other properties */

2008-11-01T13:46:37Z

Relation with other properties

← Older revision		Revision as of 13:46, 1 November 2008
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	===Stronger properties===		===Stronger properties===

	* [[Subgroup of prime index]]		* [[Weaker than::Subgroup of index two]]
	* [[Normal subgroup of finite index]]		* [[Weaker than::Subgroup of prime index]]
			* [[Weaker than::Normal subgroup of finite index]]

			===Weaker properties===

			* [[Stronger than::Subgroup of finite double coset index]]
			* [[Stronger than::Subgroup contained in finitely many intermediate subgroups]]
			* [[Stronger than::Almost normal subgroup]]
			* [[Stronger than::Nearly normal subgroup]]

	===Conjunction with other properties===		===Conjunction with other properties===

Vipul at 13:44, 1 November 2008

2008-11-01T13:44:13Z

← Older revision		Revision as of 13:44, 1 November 2008
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	{{transfercondn}}		{{transfercondn}}

	If <math>H</math> has finite index in <math>G</math> and <math>K \le G</math> is any subgroup, <math>H \cap K</math> has finite index in <math>K</math>. In fact, <math>[K:H \cap K]</math> is bounded from above by <math>[G:H]</math>. {{~~further~~\|[[Index satisfies transfer inequality]]}}		If <math>H</math> has finite index in <math>G</math> and <math>K \le G</math> is any subgroup, <math>H \cap K</math> has finite index in <math>K</math>. In fact, <math>[K:H \cap K]</math> is bounded from above by <math>[G:H]</math>. {{proofat\|[[Index satisfies transfer inequality]]}}
	{{finite-intersection-closed}}		{{finite-intersection-closed}}

	A finite intersection of subgroups of finite index again has finite index. In fact, the index of the intersection is bounded from above by the product of the indices of each subgroup. {{{proofat\|[[Index satisfies intersection inequality]]}}		A finite intersection of subgroups of finite index again has finite index. In fact, the index of the intersection is bounded from above by the product of the indices of each subgroup. {{proofat\|[[Index satisfies intersection inequality]]}}

	{{upward-closed}}		{{upward-closed}}

Vipul at 13:43, 1 November 2008

2008-11-01T13:43:49Z

← Older revision		Revision as of 13:43, 1 November 2008
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	The property of having finite index is transitive, viz a subgroup of finite index in a subgroup of finite index again has finite index. This follows essentially from the fact that the index is multiplicative. {{proofat\|[[Index is multiplicative]]}}		The property of having finite index is transitive, viz a subgroup of finite index in a subgroup of finite index again has finite index. This follows essentially from the fact that the index is multiplicative. {{proofat\|[[Index is multiplicative]]}}

			{{intsubcondn}}

			If <math>H</math> has finite index in <math>G</math> and <math>K</math> is any intermediate subgroup, the index of <math>H</math> in <math>K</math> is a factor of the index of <math>H</math> in <math>G</math>.

			{{transfercondn}}

			If <math>H</math> has finite index in <math>G</math> and <math>K \le G</math> is any subgroup, <math>H \cap K</math> has finite index in <math>K</math>. In fact, <math>[K:H \cap K]</math> is bounded from above by <math>[G:H]</math>. {{further\|[[Index satisfies transfer inequality]]}}
	{{finite-intersection-closed}}		{{finite-intersection-closed}}

	A finite intersection of subgroups of finite index again has finite index. In fact, the index of the intersection is bounded from above by the product of the indices of each subgroup. {{~~fillin~~}}		A finite intersection of subgroups of finite index again has finite index. In fact, the index of the intersection is bounded from above by the product of the indices of each subgroup. {{{proofat\|[[Index satisfies intersection inequality]]}}

	{{upward-closed}}		{{upward-closed}}
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	Any subgroup containing a subgroup of finite index is also of finite index. In fact, if <math>H</math> has finite index in <math>G</math> and <math>K</math> is any intermediate subgroup, the index of <math>K</math> in <math>G</math> is a divisor of the index of <math>H</math> in <math>G</math>.		Any subgroup containing a subgroup of finite index is also of finite index. In fact, if <math>H</math> has finite index in <math>G</math> and <math>K</math> is any intermediate subgroup, the index of <math>K</math> in <math>G</math> is a divisor of the index of <math>H</math> in <math>G</math>.

	~~{{intsubcondn}}~~

	If <math>H</math> has finite index in <math>G</math> and <math>K</math> is any intermediate subgroup, the index of <math>H</math> in <math>K</math> is a factor of the index of <math>H</math> in <math>G</math>.

	{{finite-dirprodclosedsgp}}		{{finite-dirprodclosedsgp}}

Vipul: /* Metaproperties */

2008-11-01T13:21:09Z

Metaproperties

← Older revision		Revision as of 13:21, 1 November 2008
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	{{transitive}}		{{transitive}}

	The property of having finite index is transitive, viz a subgroup of finite index in a subgroup of finite index again has finite index. This follows essentially from the fact that the index is multiplicative. {{~~fillin~~}}		The property of having finite index is transitive, viz a subgroup of finite index in a subgroup of finite index again has finite index. This follows essentially from the fact that the index is multiplicative. {{proofat\|[[Index is multiplicative]]}}

	{{finite-intersection-closed}}		{{finite-intersection-closed}}

Vipul: 8 revisions

2008-05-08T00:19:24Z

8 revisions

← Older revision	Revision as of 00:19, 8 May 2008
(No difference)

Vipul at 20:16, 19 January 2008

2008-01-19T20:16:57Z

← Older revision		Revision as of 20:16, 19 January 2008
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	{{subgroup property}}		{{subgroup property}}

			{{finitarily tautological subgroup property}}

	==Definition==		==Definition==