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On locally nilpotent groups

Published online by Cambridge University Press:  24 October 2008

D. H. McLain  and
P. Hall
D. H. McLain
Affiliation:
PeterhouseCambridge
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Extract

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 52 , Issue 1 , January 1956 , pp. 5 - 11
Copyright
Copyright © Cambridge Philosophical Society 1956

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References

REFERENCES

(1)Ado, I. D.Nilpotent algebras and p-groups. C.R. Acad. Sci. U.R.S.S. (N.S.), 40 (1943), 299–301.Google Scholar
(2)Ado, I. D.On locally finite p-groups with minimal condition for normal divisors. C.R. Acad. Sci. U.R.S.S. (N.S.), 54 (1946), 471–3.Google Scholar
(3)Ado, I. D.Proof of the countability of locally finite p-groups with minimal condition for normal divisors. Dokl. Akad. Nauk S.S.S.R. (N.S.), 58 (1947), 523–4 (Russian).Google Scholar
(4)Baer, R.Nilpotent groups and their generalisations. Trans. Amer. math. Soc. 47 (1940), 393–434.CrossRefGoogle Scholar
(5)Čarin, V. S.On the minimal condition for normal divisors of a locally soluble group. Mat. Sborn (N.S.), 33 (75) (1953), 27–36 (Russian).Google Scholar
(6)Hall, P.Finiteness conditions in soluble groups. Proc. Lond. math. Soc. (3), 4 (1954), 419–36.Google Scholar
(7)Higman, G. and Neumann, B. H.Two questions of Itô. J. Lond. math. Soc. 29 (1954), 84–8.CrossRefGoogle Scholar
(8)Jennings, S. A.A note on chain conditions in nilpotent rings and groups. Bull. Amer. math. Soc. 50 (1944), 759–63.CrossRefGoogle Scholar
(9)Kuroš, A. G. and Černikov, S. N.Soluble and nilpotent groups. Usp. mat. Nauk (N.S.), 2·3 (19) (1947), 18–59 (Russian). (English translation: Amer. math. Soc. Translation no. 80, 1953.)Google Scholar
(10)McLain, D. H.A characteristically simple group. Proc. Camb. phil. Soc. 50 (1954), 641–2.CrossRefGoogle Scholar
(11)Zassenhaus, H.Lehrbuch der Gruppentheorie (Leipzig—Berlin, 1937).Google Scholar