# locally finite group

A group $G$ is *locally finite ^{}* if any finitely generated subgroup of $G$ is finite.

A locally finite group is a torsion group^{}. The converse^{}, also known as the Burnside Problem, is not true. Burnside, however, did show that if a matrix group^{} is torsion, then it is locally finite.

(Kaplansky) If $G$ is a group such that for a normal subgroup^{} $N$ of $G$, $N$ and $G/N$ are locally finite, then $G$ is locally finite.

A solvable torsion group is locally finite. To see this, let $G={G}_{0}\supset {G}_{1}\supset \mathrm{\cdots}\supset {G}_{n}=(1)$ be a composition series^{} for $G$. We have that each ${G}_{i+1}$ is normal in ${G}_{i}$ and the factor group ${G}_{i}/{G}_{i+1}$ is abelian^{}. Because $G$ is a torsion group, so is the factor group ${G}_{i}/{G}_{i+1}$. Clearly an abelian torsion group is locally finite. By applying the fact in the previous paragraph for each step in the composition series, we see that $G$ must be locally finite.

## References

- 1 E. S. Gold and I. R. Shafarevitch, On towers of class fields, Izv. Akad. Nauk SSR, 28 (1964) 261-272.
- 2 I. N. Herstein, Noncommutative Rings, The Carus Mathematical Monographs, Number 15, (1968).
- 3 I. Kaplansky, Notes on Ring Theory, University of Chicago, Math Lecture Notes, (1965).
- 4 C. Procesi, On the Burnside problem, Journal of Algebra, 4 (1966) 421-426.

Title | locally finite group |
---|---|

Canonical name | LocallyFiniteGroup |

Date of creation | 2013-03-22 14:18:44 |

Last modified on | 2013-03-22 14:18:44 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 6 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 20F50 |

Related topic | LocallyCalP |

Related topic | PeriodicGroup |

Related topic | ProofThatLocalFinitenessIsClosedUnderExtension |

Defines | locally finite |