Group generated by periodic elements

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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

Definition

A group is termed a group generated by periodic elements or group generated by elements of finite order if it satisfies the following equvalent conditions:

  1. It has a generating set where all the elements of the generating set have finite order in the group.
  2. It is a join of finite subgroups.

Relation with other properties

Conjunction with other properties

Conjunction Other component of conjunction Additional comments
periodic abelian group abelian group For an abelian group, being generated by periodic elements is equivalent to the whole group being a periodic group, and is also equivalent to the whole group being a locally finite group.
periodic nilpotent group nilpotent group For a nilpotent group, being generated by periodic elements is equivalent to the whole group being a periodic group, and is also equivalent to the whole group being a locally finite group.
solvable group generated by periodic elements solvable group

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite group has finitely many elements Locally finite group, Periodic group|FULL LIST, MORE INFO
locally finite group every finitely generated subgroup is finite 2-locally finite group, Periodic group|FULL LIST, MORE INFO
periodic group every element has finite order |FULL LIST, MORE INFO
group generated by finitely many periodic elements has a finite generating set with every element of finite order |FULL LIST, MORE INFO