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 $G=G_{0}\supset G_{1}\supset\cdots\supset G_{n}=(1)$ 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 LocallyFiniteGroup 2013-03-22 14:18:44 2013-03-22 14:18:44 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 20F50 LocallyCalP PeriodicGroup ProofThatLocalFinitenessIsClosedUnderExtension locally finite