Periodic group: Difference between revisions

Latest revision as of 15:06, 17 April 2013

The term periodic group is also used for group with periodic cohomology

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

Definition

A group is termed a periodic group or torsion group if it satisfies the following equivalent conditions:

Every element of the group has finite 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.
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 periodic group may be finite or infinite.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes	periodicity is subgroup-closed	If $G$ is a periodic group and $H$ is a subgroup of $G$ , then $H$ is also a periodic group.
quotient-closed group property	Yes	periodicity is quotient-closed	If $G$ is a periodic group and $H$ is a normal subgroup of $G$ , then the quotient group $G / H$ is also a periodic group.
extension-closed group property	Yes	periodicity is extension-closed	If $G$ is a group and $H$ is a normal subgroup of $G$ such that both $H$ and $G / H$ are periodic groups, then $G$ is also a periodic group.
restricted direct product-closed group property	Yes	periodicity is restricted direct product-closed	Suppose $G_{i}, i \in I$ are all periodic groups, then the restricted external direct product of the $G_{i}$ s is also a periodic group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group of finite exponent	the exponent of the group is finite. This is equivalent to saying that the orders of all elements have a uniform finite bound.			\|FULL LIST, MORE INFO
finite group	the whole group is finite.			\|FULL LIST, MORE INFO
Artinian group	satisfies the descending chain condition on subgroups	Artinian implies periodic	periodic not implies Artinian	\|FULL LIST, MORE INFO
locally finite group	every finitely generated subgroup is finite	locally finite implies periodic	periodic not implies locally finite	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group having no free non-abelian subgroup				\|FULL LIST, MORE INFO
group generated by periodic elements				\|FULL LIST, MORE INFO

@@ Line 6: / Line 6: @@
 ==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 it satisfies the following equivalent conditions:
-==Relation with other properties==
+# 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.
-===Stronger properties===
+# 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]].
-* [[Bounded-exponent group]]
+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.
-* [[Finite group]]
 ==Metaproperties==
-{{S-closed}}
+{| 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.
+|}
-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.
+==Relation with other properties==
-{{Q-closed}}
+===Stronger properties===
-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.
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[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===
-More generally, an arbitrary [[restricted direct product]] of periodic groups is periodic.
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::group having no free non-abelian subgroup]] || || || || {{intermediate notions short|group having no free non-abelian subgroup|periodic group}}
+|-
+| [[Stronger than::group generated by periodic elements]] || || || || {{intermediate notions short|group generated by periodic elements|periodic group}}
+|}