Periodic group - Revision history

Vipul at 15:06, 17 April 2013

2013-04-17T15:06:25Z

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	# Every element of the group has finite [[defining ingredient::order of an element\|order]].		# Every element of the group has finite [[defining ingredient::order of an element\|order]].
	# The group is a union of finite subgroups, i.e., it is the union of a collection of subgroups, each of which is finite.		# The group is a union of finite subgroups, i.e., it is the union of a collection of subgroups, each of which is finite.
			# Every submonoid of the group (i.e., every subset that contains the identity element and is closed under multiplication, making it a [[monoid]]) is a [[subgroup]].
			# Every nonempty subsemigroup of the group (i.e., every subset that is closed under multiplication) is a [[subgroup]].

	Note that we do not assume a uniform bound on the orders of all elements. Thus, the [[exponent of a group\|exponent]] of a periodic group may be finite or infinite.		Note that we do not assume a uniform bound on the orders of all elements. Thus, the [[exponent of a group\|exponent]] of a periodic group may be finite or infinite.

Vipul at 00:12, 31 March 2013

2013-03-31T00:12:40Z

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	==Definition==		==Definition==

	A [[group]] is termed a '''periodic group''' or '''torsion group''' if ~~every~~ element of the group has finite [[defining ingredient::order of an element\|order]].		A [[group]] is termed a '''periodic group''' or '''torsion group''' if it satisfies the following equivalent conditions:

			# Every element of the group has finite [[defining ingredient::order of an element\|order]].
			# The group is a union of finite subgroups, i.e., it is the union of a collection of subgroups, each of which is finite.

	Note that we do not assume a uniform bound on the orders of all elements. Thus, the [[exponent of a group\|exponent]] of a periodic group may be finite or infinite.		Note that we do not assume a uniform bound on the orders of all elements. Thus, the [[exponent of a group\|exponent]] of a periodic group may be finite or infinite.

Vipul: /* Definition */

2013-03-30T00:21:51Z

Definition

← Older revision		Revision as of 00:21, 30 March 2013
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	==Definition==		==Definition==

	A [[group]] is termed a '''periodic group''' or '''torsion group''' if every element of the group has finite order.		A [[group]] is termed a '''periodic group''' or '''torsion group''' if every element of the group has finite [[defining ingredient::order of an element\|order]].

			Note that we do not assume a uniform bound on the orders of all elements. Thus, the [[exponent of a group\|exponent]] of a periodic group may be finite or infinite.

	==Metaproperties==		==Metaproperties==

Vipul at 00:17, 30 March 2013

2013-03-30T00:17:38Z

← Older revision		Revision as of 00:17, 30 March 2013
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	A [[group]] is termed a '''periodic group''' or '''torsion group''' if every element of the group has finite order.		A [[group]] is termed a '''periodic group''' or '''torsion group''' if every element of the group has finite order.

			==Metaproperties==

			{\| class="sortable" border="1"
			! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
			\|-
			\| [[satisfies metaproperty::subgroup-closed group property]] \|\| Yes \|\| [[periodicity is subgroup-closed]] \|\| If <math>G</math> is a periodic group and <math>H</math> is a subgroup of <math>G</math>, then <math>H</math> is also a periodic group.
			\|-
			\| [[satisfies metaproperty::quotient-closed group property]] \|\| Yes \|\| [[periodicity is quotient-closed]] \|\| If <math>G</math> is a periodic group and <math>H</math> is a [[normal subgroup]] of <math>G</math>, then the [[quotient group]] <math>G/H</math> is also a periodic group.
			\|-
			\| [[satisfies metaproperty::extension-closed group property]] \|\| Yes \|\| [[periodicity is extension-closed]] \|\| If <math>G</math> is a group and <math>H</math> is a normal subgroup of <math>G</matH> such that both <math>H</math> and <math>G/H</math> are periodic groups, then <math>G</math> is also a periodic group.
			\|-
			\| [[satisfies metaproperty::restricted direct product-closed group property]] \|\| Yes \|\| [[periodicity is restricted direct product-closed]] \|\| Suppose <math>G_i, i \in I</math> are all periodic groups, then the [[restricted external direct product]] of the <math>G_i</math>s is also a periodic group.
			\|}

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

	* [[Weaker than::~~Group~~ of finite exponent]]		{\| class="sortable" border="1"
	* [[Weaker than::~~Finite~~ group]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Weaker than::Artinian group]]~~: {{proofofstrictimplicationat~~\|[[Artinian implies periodic]]\|[[~~Periodic~~ not implies Artinian]]}}		\|-
	* [[Weaker than::~~Locally~~ finite group]]~~: {{proofofstrictimplicationat~~\|[[~~Locally~~ finite implies periodic]]\|[[~~Periodic~~ not implies locally finite]]}}		\| [[Weaker than::group of finite exponent]] \|\| the [[exponent of a group\|exponent]] of the group is finite. This is equivalent to saying that the orders of all elements have a uniform finite bound. \|\| \|\| \|\| {{intermediate notions short\|periodic group\|group of finite exponent}}
			\|-
			\| [[Weaker than::finite group]] \|\| the whole group is finite. \|\| \|\| \|\| {{intermediate notions short\|periodic group\|finite group}}
			\|-
			\| [[Weaker than::Artinian group]] \|\| satisfies the descending chain condition on subgroups \|\| [[Artinian implies periodic]]\|\| [[periodic not implies Artinian]] \|\| {{intermediate notions short\|periodic group\|Artinian group}}
			\|-
			\| [[Weaker than::locally finite group]] \|\| every finitely generated subgroup is finite \|\| [[locally finite implies periodic]] \|\| [[periodic not implies locally finite]] \|\| {{intermediate notions short\|periodic group\|locally finite group}}
			\|}
	===Weaker properties===		===Weaker properties===

	* [[Stronger than::~~Group~~ having no free non-abelian subgroup]]		{\| class="sortable" border="1"
	* [[Stronger than::~~Group~~ generated by periodic elements]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
	~~==Metaproperties==~~		\| [[Stronger than::group having no free non-abelian subgroup]] \|\| \|\| \|\| \|\| {{intermediate notions short\|group having no free non-abelian subgroup\|periodic group}}
			\|-
	{{~~S-closed}}~~		\| [[Stronger than::group generated by periodic elements]] \|\| \|\| \|\| \|\| {{intermediate notions short\|group generated by periodic elements\|periodic group}}
			\|}
	~~Any subgroup of a periodic~~ group is periodic~~. That's because the property of being~~ periodic ~~depends on a property that every individual element must satisfy, and this property doesn't depend on how big the ambient~~ group ~~is.~~

	~~{{Q-closed~~}}

	~~Any quotient of a periodic group is periodic. That's because, under a homomorphism, elements of finite order go to elements of finite order.~~

	~~{{finite-DP-closed}~~}

	~~A direct product of finitely many periodic groups is periodic. That's because, under a direct product, the order of an element is the least common multiple of the orders of each of its projections.~~

	~~More generally, an arbitrary [[restricted direct product]] of periodic groups is periodic.~~

Vipul: /* Relation with other properties */

2009-09-10T01:52:05Z

Relation with other properties

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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::~~Bounded-~~exponent ~~group~~]]		* [[Weaker than::Group of finite exponent]]
	* [[Weaker than::Finite group]]		* [[Weaker than::Finite group]]
	* [[Weaker than::Artinian group]]: {{proofofstrictimplicationat\|[[Artinian implies periodic]]\|[[Periodic not implies Artinian]]}}		* [[Weaker than::Artinian group]]: {{proofofstrictimplicationat\|[[Artinian implies periodic]]\|[[Periodic not implies Artinian]]}}

Vipul: /* Relation with other properties */

2009-03-01T23:11:31Z

Relation with other properties

← Older revision		Revision as of 23:11, 1 March 2009
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	* [[Weaker than::Finite group]]		* [[Weaker than::Finite group]]
	* [[Weaker than::Artinian group]]: {{proofofstrictimplicationat\|[[Artinian implies periodic]]\|[[Periodic not implies Artinian]]}}		* [[Weaker than::Artinian group]]: {{proofofstrictimplicationat\|[[Artinian implies periodic]]\|[[Periodic not implies Artinian]]}}
			* [[Weaker than::Locally finite group]]: {{proofofstrictimplicationat\|[[Locally finite implies periodic]]\|[[Periodic not implies locally finite]]}}

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

	* [[Stronger than::Group having no free non-abelian subgroup]]		* [[Stronger than::Group having no free non-abelian subgroup]]
			* [[Stronger than::Group generated by periodic elements]]

	==Metaproperties==		==Metaproperties==

Vipul at 23:09, 1 March 2009

2009-03-01T23:09:55Z

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	===Stronger properties===		===Stronger properties===

	* [[Bounded-exponent group]]		* [[Weaker than::Bounded-exponent group]]
	* [[Finite group]]		* [[Weaker than::Finite group]]
			* [[Weaker than::Artinian group]]: {{proofofstrictimplicationat\|[[Artinian implies periodic]]\|[[Periodic not implies Artinian]]}}

			===Weaker properties===

			* [[Stronger than::Group having no free non-abelian subgroup]]

	==Metaproperties==		==Metaproperties==

Vipul at 19:40, 9 July 2008

2008-07-09T19:40:22Z

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			{{quotation\|''The term '''periodic group''' is also used for [[group with periodic cohomology]]''}}
	{{group property}}		{{group property}}

Vipul at 14:02, 2 July 2008

2008-07-02T14:02:39Z

← Older revision		Revision as of 14:02, 2 July 2008
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	* [[Bounded-exponent group]]		* [[Bounded-exponent group]]
	* [[Finite group]]		* [[Finite group]]

			==Metaproperties==

			{{S-closed}}

			Any subgroup of a periodic group is periodic. That's because the property of being periodic depends on a property that every individual element must satisfy, and this property doesn't depend on how big the ambient group is.

			{{Q-closed}}

			Any quotient of a periodic group is periodic. That's because, under a homomorphism, elements of finite order go to elements of finite order.

			{{finite-DP-closed}}

			A direct product of finitely many periodic groups is periodic. That's because, under a direct product, the order of an element is the least common multiple of the orders of each of its projections.

			More generally, an arbitrary [[restricted direct product]] of periodic groups is periodic.

Vipul: 1 revision

2008-05-08T00:00:30Z

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