Periodic solvable group

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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 Locally finite group|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 Solvable group generated by periodic elements|FULL LIST, MORE INFO