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Some topological properties of residually Černikov groups

Published online by Cambridge University Press:  18 May 2009

M. R. Dixon
M. R. Dixon
Affiliation:
Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901, U.S.A.
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In this paper we shall indicate how to generalise the concept of a cofinite group (see [7]). We recall that any residually finite group can be made into a topological group by taking as a basis of neighbourhoods of the identity precisely the normal subgroups of finite index. The class of compact cofinite groups is then easily seen to be the class of profinite groups, where a group is profinite if and only if it is an inverse limit of finite groups. It turns out that every cofinite group can be embedded as a dense subgroup of a profinite group. This has important consequences for the class of countable locally finite-soluble groups with finite Sylow p-subgroups for all primes p, as shown in [7] and [14].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 23 , Issue 1 , January 1982 , pp. 65 - 82
Copyright
Copyright © Glasgow Mathematical Journal Trust 1982

References

REFERENCES

1.Baer, R., Lokal endlich-auflösbare Gruppen mit endlichen Sylowuntergruppen, J. Reine Angew. Math. 239/240 (1969), 109–144.Google Scholar
2.Bryant, R. M., The verbal topology of a group, J. Algebra 48 (1977), 340–346.CrossRefGoogle Scholar
3.Dixon, M. R., Formation theory in a class of locally finite groups, Ph.D. thesis, University of Warwick (1979).Google Scholar
4.Dixon, M. R. and Tomkinson, M. J., The local conjugacy of some Sylow bases in a class of locally finite groups, J. London Math. Soc. (2) 21 (1980), 225–228.CrossRefGoogle Scholar
5.Gol'berg, P. A., Sylow bases of infinite groups, Mar. Sb. 32 (1953), 465–476.Google Scholar
6.Hartley, B., Sylow subgroups of locally finite groups, Proc. London Math. Soc. (3) 23 (1971), 159–192.CrossRefGoogle Scholar
7.Hartley, B., Profinite and residually finite groups, Rocky Mountain J. Math. 7 (1977), 193–217.CrossRefGoogle Scholar
8.Higgins, P., An introduction to topological groups, London Math. Soc. Lecture Notes Series 15 (Cambridge University Press, 1974).Google Scholar
9.Kegel, O. H. and Wehrfritz, B. A. F., Locally finite groups (North-Holland, 1973).Google Scholar
10.Kovács, L. G., Neumann, B. H. and Vries, H. De, Some Sylow subgroups, Proc. Roy. Soc. Ser. A 260 (1961), 304–316.Google Scholar
11.Kuroš, A. G., Theory of groups, Vol. 2 (Chelsea, 1956).Google Scholar
12.Massey, N., Locally finite groups with min-p for all p. I, J. London Math. Soc. (2) 12 (1975), 7–14.CrossRefGoogle Scholar
13.Neumann, B. H., Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236–248.CrossRefGoogle Scholar
14.Parker, J. R., A topological approach to a class of residually finite groups, Ph.D. thesis, University of Warwick (1973).Google Scholar
15.Simmons, G. F., Introduction to topology and modem analysis (McGraw-Hill, 1963).Google Scholar
16.Stewart, I., Conjugacy theorems for a class of locally finite Lie algebras, Compositio Math. 30 (1975), 181–210.Google Scholar