# Equivalence of definitions of periodic solvable group

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

## Statement

The following are equivalent for a group $G$:

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.

## Facts used

1. Locally finite implies periodic
2. Periodicity is subgroup-closed
3. Periodicity is quotient-closed
4. Equivalence of definitions of periodic abelian group: This states that for an abelian group, being periodic is equivalent to being locally finite.
5. 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 $G$ are locally finite. Thus, by Fact (5), $G$ itself is locally finite.