Periodic solvable group

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

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: locally finite group and solvable group
View other group property conjunctions OR view all group properties

Definition

A group is termed a periodic solvable group or locally finite solvable group if it satisfies the following equivalent conditions:

1. It is both a locally finite group (every finitely generated subgroup is finite) and a solvable group (the derived series reaches the trivial subgroup in finitely many steps).
2. It is both a periodic group (every element has finite order) and a solvable group (the derived series reaches the trivial subgroup in finitely many steps).
3. It is a solvable group and all the groups obtained by taking quotients of successive members of its derived series are periodic abelian groups.

Equivalence of definitions

Further information: equivalence of definitions of periodic solvable group

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
periodic abelian group				|FULL LIST, MORE INFO
periodic nilpotent group				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
locally finite group	every finitely generated subgroup is finite			|FULL LIST, MORE INFO
periodic group	every element has finite order			|FULL LIST, MORE INFO
solvable group generated by periodic elements	solvable, and it has a generating set comprising periodic elements		solvable and generated by periodic elements not implies periodic	|FULL LIST, MORE INFO
solvable group				|FULL LIST, MORE INFO