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.

Bibliography

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I. N. Herstein, Noncommutative Rings, The Carus Mathematical Monographs, Number 15, (1968).

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I. Kaplansky, Notes on Ring Theory, University of Chicago, Math Lecture Notes, (1965).

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C. Procesi, On the Burnside problem, Journal of Algebra, 4 (1966) 421-426.