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 of odd exponent on generators.
Some easy results on periodic groups:
Every subgroup (http://planetmath.org/Subgroup) of a periodic group is periodic.
Every quotient (http://planetmath.org/QuotientGroup) of a periodic group is periodic.
Every extension (http://planetmath.org/GroupExtension) of a periodic group by a periodic group is periodic.
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 is not periodic, as the element that is in every coordinate has infinite order.
Some further results on periodic groups:
Every solvable periodic group is locally finite.
|Date of creation||2013-03-22 15:35:50|
|Last modified on||2013-03-22 15:35:50|
|Last modified by||yark (2760)|