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locally finite group (Definition)

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

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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See Also: locally $\cal P$, periodic group, proof that local finiteness is closed under extension

Also defines:  locally finite

Attachments:
proof that local finiteness is closed under extension (Proof) by yark
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Cross-references: abelian, factor group, composition series, solvable, normal subgroup, torsion, matrix group, converse, torsion group, finite, finitely generated subgroup, group
There are 8 references to this entry.

This is version 3 of locally finite group, born on 2004-04-16, modified 2004-12-10.
Object id is 5776, canonical name is LocallyFiniteGroup.
Accessed 3911 times total.

Classification:
AMS MSC20F50 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Periodic groups; locally finite groups)

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