locally finite groupLocallyFiniteGroup2004-04-16 22:33:112004-12-10 11:50:47Definitionlocally finite% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA group $G$ is \emph{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.
\begin{thebibliography}{9}
\bibitem{golod} E. S. Gold and I. R. Shafarevitch, {\em On towers of class fields}, Izv. Akad. Nauk SSR, 28 (1964) 261-272.
\bibitem{herstein} I. N. Herstein, {\em Noncommutative Rings}, The Carus Mathematical Monographs, Number 15, (1968).
\bibitem{kaplansky} I. Kaplansky, {\em Notes on Ring Theory}, University of Chicago, Math Lecture Notes, (1965).
\bibitem{procesi} C. Procesi, {\em On the Burnside problem}, Journal of Algebra, 4 (1966) 421-426.
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