Group generated by finitely many periodic elements

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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and group generated by periodic elements
View other group property conjunctions OR view all group properties

Definition

A group is termed a group generated by finitely many periodic elements if it satisfies the following equivalent conditions:

  1. It has a generating set that is finite and comprises elements only of finite orders.
  2. It is both a finitely generated group and a group generated by periodic elements (this definition is a priori different from the first one in that we are not assuming that the finite generating set and the generating set comprising periodic elements are the same).
  3. It is a join of finitely many finite subgroups.
  4. It is a join of finitely many finite cyclic subgroups.

Equivalence of definitions

Further information: equivalence of definition of group generated by finitely many periodic elements

Relation with other properties

Conjunction with other properties

Conjunction Other component of conjunction Comments
finite abelian group abelian group See also equivalence of definitions of periodic abelian group: this shows that for an abelian group, being generated by periodic elements is equivalent to being locally finite, which in turn means that if it is also finitely generated, it is finite.
finite nilpotent group nilpotent group See also equivalence of definitions of periodic nilpotent group: this shows that for a nilpotent group, being generated by periodic elements is equivalent to being locally finite, which in turn means that if it is also finitely generated, it is finite.
solvable group generated by finitely many periodic elements solvable group