Classes of topological groups suggested by Galois theory. (English) Zbl 0742.20029
Let \(G\) be a profinite abelian group. Motivated by the Galois theory of infinite extensions, cf. H. Bass and the author [J. Indian Math. Soc., New Ser. 36, 1-7 (1972; Zbl 0284.20037)], one is interested in dense totally bounded subgroups \(H\) of \(G\) such that all subgroups of \(H\) are closed in \(H\). The author proves that \(G\) and \(H\) satisfy these conditions if and only if \(G\) is the profinite completion of \(H\), and \(H\) is an extension of a free abelian group of finite rank by a torsion group with primary parts of finite exponents.
The second part of the paper contains various general results on topologically complete groups, i.e. topological groups with trivial center such that every topological automorphism is an inner automorphism.
The structure of profinite abelian groups which admit only one compact group topology has been determined by A. Hulanicki [Diss. Math. (Rozprawy Mat.) 38, 1–58 (1964; Zbl 0119.03301)]. In theorem 3.1, the author describes all locally compact group topologies on these groups. Finally, he extends this description to (possibly nonabelian) profinite groups with Sylow subgroups of similar shape.
The second part of the paper contains various general results on topologically complete groups, i.e. topological groups with trivial center such that every topological automorphism is an inner automorphism.
The structure of profinite abelian groups which admit only one compact group topology has been determined by A. Hulanicki [Diss. Math. (Rozprawy Mat.) 38, 1–58 (1964; Zbl 0119.03301)]. In theorem 3.1, the author describes all locally compact group topologies on these groups. Finally, he extends this description to (possibly nonabelian) profinite groups with Sylow subgroups of similar shape.
Reviewer: Th.Grundhöfer (Tübingen)
MSC:
| 20E18 | Limits, profinite groups |
| 20K45 | Topological methods for abelian groups |
| 22B05 | General properties and structure of LCA groups |
| 22A05 | Structure of general topological groups |
| 12F12 | Inverse Galois theory |
| 22C05 | Compact groups |
Keywords:
dense totally bounded subgroups; profinite completion; free abelian group of finite rank; topologically complete groups; inner automorphism; profinite abelian groups; compact group topology; locally compact group topologies; Sylow subgroupsReferences:
| [1] | Baer, R., Absolute retracts in group theory, Bull. Amer. Math. Soc., 52, 501-506 (1946) · Zbl 0061.02702 |
| [2] | Bass, H.; Soundararajan, T., A property of profinite groups and the converse of classical Galois theory, J. Indian Math. Soc., 36, 1-7 (1972) · Zbl 0284.20037 |
| [3] | Comfort, W. W., Topological groups, (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), Elsevier: Elsevier Amsterdam), 1143-1263 · Zbl 0604.22002 |
| [4] | Comfort, W. W.; Ross, K. A., Topologies induced by groups of characters, Fund. Math., 55, 283-291 (1964) · Zbl 0138.02905 |
| [5] | Comfort, W. W.; Ross, K. A., Pseudocompactness and uniform continuity in topological groups, Pacific J. Math., 16, 483-496 (1966) · Zbl 0214.28502 |
| [6] | Comfort, W. W.; Saks, V., Countably compact groups and finest totally bounded topologies, Pacific J. Math., 49, 33-44 (1973) · Zbl 0271.22001 |
| [7] | Corwin, L., Uniqueness of the topology for the \(p\)-adic integers, Proc. Amer. Math. Soc., 55, 432-434 (1976) · Zbl 0323.22005 |
| [8] | de Vries, J., Pseudo compactness and the Stone-Čech compactification for topological groups, Nieuw Archief voor Wiskunde, 23, 35-48 (1975) · Zbl 0296.22003 |
| [9] | Fuchs, L., Infinite Abelian Groups, I (1970), Academic Press: Academic Press New York · Zbl 0209.05503 |
| [10] | Fuchs, L., Infinite Abelian groups, II (1973), Academic Press: Academic Press New York · Zbl 0257.20035 |
| [11] | Halmos, P. R., Comment on the real line, Bull. Amer. Math. Soc., 50, 877-878 (1944) · Zbl 0061.04404 |
| [12] | Hewitt, E., A remark on characters of locally compact Abelian groups, Fund. Math., 53, 55-64 (1963) · Zbl 0123.02904 |
| [13] | Hewitt, E.; Ross, K. A., Abstract Harmonic Analysis, Vol. I (1963), Springer: Springer Berlin · Zbl 0115.10603 |
| [14] | Hochschild, G., The Structure of Lie Groups (1965), Holden-Day: Holden-Day San Francisco, CA · Zbl 0131.02702 |
| [15] | Hulanicki, A., Compact Abelian groups and extensions of Haar measure, Rozprawy Mat., 38, 1-58 (1964) · Zbl 0119.03301 |
| [16] | Ikeda, M., Completeness of the absolute Galois group of the algebraic closure of the rational number field, J. Reine Angew. Math., 2981, 1-22 (1977) · Zbl 0366.12008 |
| [17] | Kallman, R. R., A uniqueness result for topological groups, Proc. Amer. Math. Soc., 54, 439-440 (1976) · Zbl 0317.22001 |
| [18] | Krull, W., Galoissche theorie der unendlichen algebraischen Erweiterungen, Math. Ann., 100, 687-698 (1928) · JFM 54.0157.02 |
| [19] | Leptin, H., Ein Darstellungssatz für kompakte total unzusammenhängende Gruppen, Arch. Math., 6, 371-373 (1955) · Zbl 0057.26404 |
| [20] | Montgomery, D.; Zippin, L., Topological Transformation Groups (1955), Interscience: Interscience New York · JFM 66.0959.03 |
| [21] | Mycielski, J., Some properties of connected compact groups, Colloq. Math., 5, 162-166 (1958) · Zbl 0088.02802 |
| [22] | Neukirch, J., Kennzeichnung der \(p\)-adischen und der endlichen algebraischen Zahlkörper, Invent. Math., 6, 296-314 (1969) · Zbl 0192.40102 |
| [23] | Pletch, A., Strong completeness in profinite groups, Pacific J. Math., 97, 203-208 (1981) · Zbl 0451.20027 |
| [24] | Rajagopalan, M., Topologies in locally compact groups, Math. Ann., 176, 169-180 (1968) · Zbl 0174.05601 |
| [25] | Rickert, N. W., Locally compact topologies for groups, Trans. Amer. Math. Soc., 126, 225-235 (1967) · Zbl 0189.32404 |
| [26] | Ross, K. A., Closed subgroups of locally compact Abelian groups, Fund. Math., 56, 241-244 (1965) · Zbl 0132.27701 |
| [27] | Schreier, J.; Ulam, S., Über die Automorphismen der Permutationsgruppe der natürlichen Zahlenfolge, Fund. Math., 28, 258-260 (1937) · JFM 63.0073.01 |
| [28] | Soundararajan, T., The topological group of the \(p\)-adic integers, Publ. Math. Debrecen, 16, 75-78 (1969) · Zbl 0209.06002 |
| [29] | Uchida, K., Isomorphisms of Galois groups, J. Math. Soc. Japan, 28, 617-620 (1976) · Zbl 0329.12013 |
| [30] | Yu, Y. K., Topologically semisimple groups, Proc. London Math. Soc., 33, 515-534 (1976) · Zbl 0351.22001 |
| [31] | Zassenhaus, H. J., The Theory of Groups (1958), Chelsea, New York · Zbl 0083.24517 |
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