Finite abelian group - Revision history

Vipul at 03:59, 16 April 2017

2017-04-16T03:59:56Z

← Older revision		Revision as of 03:59, 16 April 2017
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	{{group property conjunction see examples\|finite group\|abelian group}}		{{group property conjunction see examples\|finite group\|abelian group}}

			==Metaproperties==

			{\| class="sortable" border="1"
			! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
			\|-
			\| [[satisfies metaproperty::subgroup-closed group property]] \|\| Yes \|\| follows from [[abelianness is subgroup-closed]] \|\| If <math>G</math> is a finite abelian group and <math>H</math> is a subgroup of <math>G</math>, then <math>H</math> is also a finite abelian group.
			\|-
			\| [[satisfies metaproperty::quotient-closed group property]] \|\| Yes \|\| follows from [[abelianness is quotient-closed]] \|\| If <math>G</math> is a finite abelian group and <math>H</math> is a normal subgroup of <math>G</math>, then the quotient group <math>G/H</math> is also a finite abelian group.
			\|-
			\| [[satisfies metaproperty::finite direct product-closed group property]] Yes \|\| follows from [[abelianness is direct product-closed]] \|\| If <math>G_1, G_2, \dots, G_n</math> are finite abelian groups, so is the [[external direct product]] <math>G_1 \times G_2 \times \dots \times G_n</math>.
			\|-
			\| [[satisfies metaproperty::lattice-determined group property]] \|\| No \|\| [[there exists an abelian group of prime power order that is lattice-isomorphic to a non-abelian group not of prime power order]] \|\| There exist groups <math>G_1, G_2</math> with isomorphic lattices of subgroups such that <math>G_1</math> is finite abelian and <math>G_2</math> is not.
			\|}
	==Relation with other properties==		==Relation with other properties==

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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::~~Abelian~~ group of prime power order]] \|\| \|\| \|\| \|\|		\| [[Weaker than::abelian group of prime power order]] \|\| \|\| \|\| \|\|
	\|-		\|-
	\| [[Weaker than::~~Finite~~ cyclic group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finite abelian group\|finite cyclic group}}		\| [[Weaker than::finite cyclic group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finite abelian group\|finite cyclic group}}
	\|-		\|-
	\| [[Weaker than::~~Odd~~-order abelian group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finite abelian group\|odd-order abelian group}}		\| [[Weaker than::odd-order abelian group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finite abelian group\|odd-order abelian group}}
	\|}		\|}

Vipul at 03:23, 17 December 2011

2011-12-17T03:23:01Z

← Older revision		Revision as of 03:23, 17 December 2011
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	{{group property conjunction\|finite group\|abelian group}}		{{group property conjunction\|finite group\|abelian group}}
			[[importance rank::1\| ]]
	==Definition==		==Definition==

Vipul: /* Relation with other properties */

2010-05-31T23:18:43Z

Relation with other properties

← Older revision		Revision as of 23:18, 31 May 2010
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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::Abelian group of prime power order]]		{\| class="sortable" border="1"
	* [[Weaker than::Finite cyclic group]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Weaker than::Odd-order abelian group]]		\|-
			\| [[Weaker than::Abelian group of prime power order]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Weaker than::Finite cyclic group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finite abelian group\|finite cyclic group}}
			\|-
			\| [[Weaker than::Odd-order abelian group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finite abelian group\|odd-order abelian group}}
			\|}

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

	* [[Stronger than::Finitely generated abelian group]]		{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::Finitely generated abelian group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finitely generated abelian group\|finite abelian group}}
			\|-
			\| [[Stronger than::Periodic abelian group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|periodic abelian group\|finite abelian group}}
			\|-
			\| [[Stronger than::Finite nilpotent group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finite nilpotent group\|finite abelian group}}
			\|-
			\| [[Stronger than::Finite group that is 1-isomorphic to an abelian group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finite group that is 1-isomorphic to an abelian group\|finite abelian group}}
			\|-
			\| [[Stronger than::Finite group that is order statistics-equivalent to an abelian group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finite group that is order statistics-equivalent to an abelian group\|finite abelian group}}
			\|}

Vipul at 23:40, 9 September 2009

2009-09-09T23:40:06Z

← Older revision		Revision as of 23:40, 9 September 2009
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	{{proofat\|[[Structure theorem for finitely generated abelian groups]]}}		{{proofat\|[[Structure theorem for finitely generated abelian groups]]}}

			==Examples==

			{{group property conjunction see examples\|finite group\|abelian group}}

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

Vipul: Finite Abelian group moved to Finite abelian group

2009-01-21T23:19:45Z

Finite Abelian group moved to Finite abelian group

← Older revision	Revision as of 23:19, 21 January 2009
(No difference)

Vipul: /* Symbol-free definition */

2009-01-21T23:19:23Z

Symbol-free definition

← Older revision		Revision as of 23:19, 21 January 2009
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	# It is both [[finite group\|finite]] and [[abelian group\|abelian]].		# It is both [[finite group\|finite]] and [[abelian group\|abelian]].
	# It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s.		# It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s.
	# It is isomorphic to a direct product of [[abelian group of prime power order\|~~Abelian~~ groups of prime power order]].		# It is isomorphic to a direct product of [[abelian group of prime power order\|abelian groups of prime power order]].
	# It is isomorphic to a direct product of [[cyclic group of prime power order\|cyclic groups of prime power order]].		# It is isomorphic to a direct product of [[cyclic group of prime power order\|cyclic groups of prime power order]].

Vipul: /* Symbol-free definition */

2009-01-21T23:19:14Z

Symbol-free definition

← Older revision		Revision as of 23:19, 21 January 2009
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	# It is both [[finite group\|finite]] and [[abelian group\|abelian]].		# It is both [[finite group\|finite]] and [[abelian group\|abelian]].
	# It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s.		# It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s.
	# It is isomorphic to a direct product of [[~~Abelian~~ group of prime power order\|Abelian groups of prime power order]].		# It is isomorphic to a direct product of [[abelian group of prime power order\|Abelian groups of prime power order]].
	# It is isomorphic to a direct product of [[cyclic group of prime power order\|cyclic groups of prime power order]].		# It is isomorphic to a direct product of [[cyclic group of prime power order\|cyclic groups of prime power order]].

Vipul at 23:18, 21 January 2009

2009-01-21T23:18:59Z

← Older revision		Revision as of 23:18, 21 January 2009
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	{{group property conjunction\|finite group\|~~Abelian~~ group}}		{{group property conjunction\|finite group\|abelian group}}

	==Definition==		==Definition==
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	===Symbol-free definition===		===Symbol-free definition===

	A '''finite ~~Abelian~~ group''' is a [[group]] satisfying the following equivalent conditions:		A '''finite abelian group''' is a [[group]] satisfying the following equivalent conditions:

	# It is both [[finite group\|finite]] and [[~~Abelian~~ group\|~~Abelian~~]].		# It is both [[finite group\|finite]] and [[abelian group\|abelian]].
	# It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s.		# It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s.
	# It is isomorphic to a direct product of [[Abelian group of prime power order\|Abelian groups of prime power order]].		# It is isomorphic to a direct product of [[Abelian group of prime power order\|Abelian groups of prime power order]].
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	===Equivalence of definitions===		===Equivalence of definitions===

	{{proofat\|[[Structure theorem for finitely generated ~~Abelian~~ groups]]}}		{{proofat\|[[Structure theorem for finitely generated abelian groups]]}}

	==Relation with other properties==		==Relation with other properties==
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	* [[Weaker than::Abelian group of prime power order]]		* [[Weaker than::Abelian group of prime power order]]
	* [[Weaker than::Finite cyclic group]]		* [[Weaker than::Finite cyclic group]]
	* [[Weaker than::Odd-order ~~Abelian~~ group]]		* [[Weaker than::Odd-order abelian group]]

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

	* [[Stronger than::Finitely generated ~~Abelian~~ group]]		* [[Stronger than::Finitely generated abelian group]]

Vipul: New page: {{group property conjunction|finite group|Abelian group}} ==Definition== ===Symbol-free definition=== A '''finite Abelian group''' is a group satisfying the following equivalent con...

2009-01-04T20:25:06Z

New page: {{group property conjunction|finite group|Abelian group}} ==Definition== ===Symbol-free definition=== A '''finite Abelian group''' is a group satisfying the following equivalent con...

New page

{{group property conjunction|finite group|Abelian group}}

==Definition==

===Symbol-free definition===

A '''finite Abelian group''' is a [[group]] satisfying the following equivalent conditions:

# It is both [[finite group|finite]] and [[Abelian group|Abelian]].
# It is isomorphic to a [[direct product]] of finitely many [[finite cyclic group]]s.
# It is isomorphic to a direct product of [[Abelian group of prime power order|Abelian groups of prime power order]].
# It is isomorphic to a direct product of [[cyclic group of prime power order|cyclic groups of prime power order]].

===Equivalence of definitions===

{{proofat|[[Structure theorem for finitely generated Abelian groups]]}}

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Abelian group of prime power order]]
* [[Weaker than::Finite cyclic group]]
* [[Weaker than::Odd-order Abelian group]]

===Weaker properties===

* [[Stronger than::Finitely generated Abelian group]]