# periodic group

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 $n\ge 665$ on $m>1$ generators^{}.

Some easy results on periodic groups:

###### Theorem 1.

Every subgroup^{} (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 extension^{} (http://planetmath.org/GroupExtension) 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 (http://planetmath.org/PGroup4) 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 |