# periodic group

A group $G$ is said to be periodic (or torsion) if every element of $G$ is of finite order.

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 (http://planetmath.org/Subgroup) of a periodic group is periodic.

• ###### Theorem 2.
• Every quotient (http://planetmath.org/QuotientGroup) of a periodic group is periodic.

• ###### Theorem 3.
• Every extension (http://planetmath.org/GroupExtension) 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 (http://planetmath.org/PGroup4) over all primes $p$.

Title periodic group PeriodicGroup 2013-03-22 15:35:50 2013-03-22 15:35:50 yark (2760) yark (2760) 11 yark (2760) Definition msc 20F50 torsion group LocallyFiniteGroup Torsion3 periodic torsion