periodic group

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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 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 $p$ .

Defines:

periodic, torsion

Synonym:

torsion group

Type of Math Object:

Definition

Major Section:

Reference

Mathematics Subject Classification

20F50 Periodic groups; locally finite groups

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periodic group

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periodic group

Theorem 1.

Theorem 2.

Theorem 3.

Theorem 4.

Theorem 5.

Theorem 6.

Mathematics Subject Classification

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periodic group

Primary tabs

periodic group

Theorem 1.

Theorem 2.

Theorem 3.

Theorem 4.

Theorem 5.

Theorem 6.

Mathematics Subject Classification

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