Solvable group generated by periodic elements
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) | |FULL LIST, MORE INFO |
| periodic abelian group | abelian and every element has finite order | (via periodic nilpotent) | (via periodic nilpotent) | |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 |