Equivalence of definitions of periodic solvable group

From Groupprops
Jump to: navigation, search
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.