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.