Solvable group generated by 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: solvable group and group generated by periodic elements
View other group property conjunctions OR view all group properties

Definition

A solvable group generated by periodic elements is a group that is both a solvable group and is a group generated by periodic elements, i.e., it has a generating set in which all the elements have finite order.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
solvable group generated by finitely many periodic elements any countable restricted direct power of a nontrivial finite solvable group
periodic solvable group solvable and every element has finite order (obvious) solvable and generated by periodic elements not implies periodic |FULL LIST, MORE INFO
periodic nilpotent group nilpotent and every element has finite order (via periodic solvable) (via periodic solvable) Periodic solvable group|FULL LIST, MORE INFO
periodic abelian group abelian and every element has finite order (via periodic nilpotent) (via periodic nilpotent) Periodic solvable group|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
solvable group |FULL LIST, MORE INFO
group generated by periodic elements |FULL LIST, MORE INFO