locally finite group

A group G is locally finitePlanetmathPlanetmathPlanetmath if any finitely generated subgroup of G is finite.

A locally finite group is a torsion groupPlanetmathPlanetmath. The converseMathworldPlanetmath, also known as the Burnside Problem, is not true. Burnside, however, did show that if a matrix groupMathworldPlanetmath is torsion, then it is locally finite.

(Kaplansky) If G is a group such that for a normal subgroupMathworldPlanetmath 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=G0G1Gn=(1) be a composition seriesMathworldPlanetmathPlanetmathPlanetmath for G. We have that each Gi+1 is normal in Gi and the factor group Gi/Gi+1 is abelianMathworldPlanetmath. Because G is a torsion group, so is the factor group Gi/Gi+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.


  • 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