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单词 PeriodicGroup
释义

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 (http://planetmath.org/Subgroup) of a periodic group is periodic.

  • Theorem 2.
  • Every quotient (http://planetmath.org/QuotientGroup) of a periodic group is periodic.

  • Theorem 3.
  • Every extensionPlanetmathPlanetmathPlanetmath (http://planetmath.org/GroupExtension) 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 (http://planetmath.org/PGroup4) over all primes p.

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    更新时间:2025/5/3 14:58:51