periodic group

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

All finite groupsMathworldPlanetmath 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 n665 on m>1 generatorsPlanetmathPlanetmathPlanetmath.

Some easy results on periodic groups:

Theorem 1.
  • Every subgroupMathworldPlanetmathPlanetmath ( of a periodic group is periodic.

  • Theorem 2.
  • Every quotient ( of a periodic group is periodic.

  • Theorem 3.
  • Every extensionPlanetmathPlanetmathPlanetmath ( of a periodic group by a periodic group is periodic.

  • Theorem 4.
  • Every restricted direct productPlanetmathPlanetmath of periodic groups is periodic.

  • Note that (unrestricted) direct productsMathworldPlanetmathPlanetmathPlanetmathPlanetmath of periodic groups are not necessarily periodic. For example, the direct product of all finite cyclic groupsMathworldPlanetmath /n 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 groupMathworldPlanetmath is the direct sumMathworldPlanetmath of its maximal p-groups ( over all primes p.

    Title periodic group
    Canonical name PeriodicGroup
    Date of creation 2013-03-22 15:35:50
    Last modified on 2013-03-22 15:35:50
    Owner yark (2760)
    Last modified by yark (2760)
    Numerical id 11
    Author yark (2760)
    Entry type Definition
    Classification msc 20F50
    Synonym torsion group
    Related topic LocallyFiniteGroup
    Related topic Torsion3
    Defines periodic
    Defines torsion