Periodic group
From Groupprops
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
View a complete list of group properties
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) |
Contents
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 ![]() ![]() ![]() ![]() |
quotient-closed group property | Yes | periodicity is quotient-closed | If ![]() ![]() ![]() ![]() |
extension-closed group property | Yes | periodicity is extension-closed | If ![]() ![]() ![]() ![]() ![]() ![]() |
restricted direct product-closed group property | Yes | periodicity is restricted direct product-closed | Suppose ![]() ![]() |
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. | Artinian group, Finitely generated periodic group, Finitely presented periodic group, Group of finite exponent, Locally finite group|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 | 2-locally finite group|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 |