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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 $ \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$.



"periodic group" is owned by yark.
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See Also: locally finite group, torsion

Other names:  torsion group
Also defines:  periodic, torsion
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Cross-references: primes, maximal, direct sum, abelian group, solvable, infinite order, cyclic groups, direct products, restricted direct product, generators, exponent, odd, Burnside groups, Tarski groups, locally finite, locally finite groups, finite groups, order, finite, group
There are 31 references to this entry.

This is version 8 of periodic group, born on 2005-12-01, modified 2007-07-25.
Object id is 7511, canonical name is PeriodicGroup.
Accessed 4790 times total.

Classification:
AMS MSC: 20F50 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Periodic groups; locally finite groups)
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