PlanetMath: periodic group

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periodic group

(Definition)

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 $n\ge665$ 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 $\Z/n\Z$ 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 $p$ .

"periodic group" is owned by yark.

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