## You are here

Homeperiodic group

## Primary tabs

# periodic group

All finite groups are periodic. More generally, all locally finite groups are periodic. Examples of periodic groups that are not locally finite include Tarski groups, and Burnside groups $B(m,n)$ of odd exponent $n\geq 665$ on $m>1$ generators.

Some easy results on periodic groups:

###### Theorem 1.

Every subgroup of a periodic group is periodic.

###### Theorem 2.

Every quotient of a periodic group is periodic.

###### Theorem 3.

Every extension of a periodic group by a periodic group is periodic.

###### Theorem 4.

Every restricted direct product of periodic groups is periodic.

Note that (unrestricted) direct products of periodic groups are not necessarily periodic. For example, the direct product of all finite cyclic groups $\mathbb{Z}/n\mathbb{Z}$ is not periodic, as the element that is $1$ in every coordinate has infinite order.

Some further results on periodic groups:

###### Theorem 5.

Every solvable periodic group is locally finite.

###### Theorem 6.

Every periodic abelian group is the direct sum of its maximal $p$-groups over all primes $p$.

## Mathematics Subject Classification

20F50*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith