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A group  is locally finite if any finitely generated subgroup of  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  is a group such that for a normal subgroup  of  ,  and  are locally finite, then  is locally finite.
A solvable torsion group is locally finite. To see this, let
 be a composition series for  . We have that each  is normal in  and the factor group
 is abelian. Because  is a torsion group, so is the factor group
 . 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  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.
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