Locally finite group

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A group in which every finitely-generated subgroup is finite. Any locally finite group is a torsion group (cf. Periodic group), but not conversely (see Burnside problem). An extension of a locally finite group by a locally finite group is again a locally finite group. Every locally finite group with the minimum condition for subgroups (and even for Abelian subgroups) has an Abelian subgroup of finite index [3] (see Group with a finiteness condition). A locally finite group whose Abelian subgroups have finite rank (cf. Rank of a group) has itself finite rank and contains a locally solvable subgroup (cf. Locally solvable group) of finite index.


[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2] S.N. Chernikov, "Finiteness conditions in general group theory" Uspekhi Mat. Nauk , 14 : 5 (1959) pp. 45–96 (In Russian)
[3] V.P. Shunkov, "On locally finite groups with a minimality condition for Abelian subgroups" Algebra and Logic , 9 : 5 (1970) pp. 350–370 Algebra i Logika , 9 : 5 (1970) pp. 579–615
[4] V.P. Shunkov, "On locally finite groups of finite rank" Algebra and Logic , 10 : 2 (1971) pp. 127–142 Algebra i Logika , 10 : 2 (1971) pp. 199–225
How to Cite This Entry:
Locally finite group. A.L. Shmel'kin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098