locally finite group (original) (raw)

(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=G0⊃G1⊃⋯⊃Gn=(1) be a composition series for G. We have that each Gi+1 is normal in Gi and the factor group Gi/Gi+1 is abelian. Because G is a torsion group, so is the factor group Gi/Gi+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.