# Equivalence of definitions of periodic solvable group

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term periodic solvable group

View a complete list of pages giving proofs of equivalence of definitions

## Contents

## Statement

The following are equivalent for a group :

- 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).
- 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).
- It is a solvable group and all the groups obtained by taking quotients of successive members of its derived series are periodic abelian groups.

## Facts used

- Locally finite implies periodic
- Periodicity is subgroup-closed
- Periodicity is quotient-closed
- Equivalence of definitions of periodic abelian group: This states that for an abelian group, being periodic is equivalent to being locally finite.
- Local finiteness is extension-closed

## Proof

### (1) implies (2)

This follows from Fact (1).

### (2) implies (3)

This follows from Facts (2) and (3).

### (3) implies (1)

By Fact (4), all quotients between successive members of the derived series of are locally finite. Thus, by Fact (5), itself is locally finite.