Geodesic
On Infinite Locally Finite Groups
Canadian mathematical bulletin, Tome 37 (1994) no. 4, pp. 537-544

Voir la notice de l'article provenant de la source Cambridge University Press

If G is a group such that every infinite subset of G contains a commuting pair of elements then G is centre-by-finite. This result is due to B. H. Neumann. From this it can be shown that if G is infinite and such that for every pair X, Y of infinite subsets of G there is some x in X and some y in Y that commute, then G is abelian. It is natural to ask if results of this type would hold with other properties replacing commutativity. It may well be that group axioms are restrictive enough to provide meaningful affirmative results for most of the properties. We prove the following result which is of similar nature.If G is a group such that for each positive integer n and for every n infinite subset X1,...,Xn of G there exist elements xi of Xii = 1,... ,n, such that the subgroup generated by {x1 ,... ,xn } is finite, then G is locally finite.
DOI : 10.4153/CMB-1994-078-5
Mots-clés : 20F50 
Rhemtulla, Akbar; Smith, Howard. On Infinite Locally Finite Groups. Canadian mathematical bulletin, Tome 37 (1994) no. 4, pp. 537-544. doi: 10.4153/CMB-1994-078-5
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[1] 1. Hartley, B., A general Brauer-Fowler Theorem and centralizers in locally finite groups, Pacific J. Math. 152(1992), 101–117. Google Scholar

[2] 2. Huppert, B. and Blackburn, N., Finite Groups III, Grundlehren Math. Wiss. Springer-Verlag, Berlin, Heidelberg, New York, 1982. Google Scholar

[3] 3. Kegel, O. H. and Wehrfritz, B. A. F., Locally Finite Groups, North Holland, American Elsevier, Amsterdam, London, New York, 1973. Google Scholar

[4] 4. Kim, P. S., Rhemtulla, A. and Smith, H., A characterization of infinite metabelian groups, Houston J. Math. 17(1991), 429–437. Google Scholar

[5] 5. Neumann, B. H., On a problem of Paul Erdos on groups, J. Austral. Math. Soc. 21(1976), 467–472. Google Scholar

[6] 6. Suzuki, M., Group Theory II, Grundlehren Math. Wiss. Springer-Verlag, Berlin, Heidelberg, New York, 1986. Google Scholar

[7] 7. Weisner, L., Groups in which the normalizer of every element except identity is abelian, Bull. Amer. Math. Soc. 31(1925), 413–416. Google Scholar

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