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Totally disconnected, nilpotent, locally compact groups. (English) Zbl 0893.22001

It is an old result of D. van Dantzig and E. R. van Kampen that in a locally compact, totally disconnected group, the set of compact-open subgroups is a local base at the identity. Examples show, however, that the set of open, normal subgroups may fail to be a local basis. The present author now shows, improving work of Y. Guivarch, M. Keane and B. Roynette [Marches aléatoires sur les groupes de Lie, Lecture Notes in Math. 624. Berlin etc. (1977; Zbl 0367.60081)], that in a locally compact, compactly generated, totally disconnected nilpotent group, the set of compact-open normal subgroups is a local base at the identity. An example shows further that the hypothesis “compactly generated” cannot be omitted.

MSC:

22A05 Structure of general topological groups
22D05 General properties and structure of locally compact groups
20F18 Nilpotent groups
54H11 Topological groups (topological aspects)

Citations:

Zbl 0367.60081
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References:

[1] Montgomery, Topological transformation groups (1955) · Zbl 0068.01904
[2] Hewitt, Abstract harmonic analysis (1963) · Zbl 0089.10806 · doi:10.1007/978-3-662-00102-8
[3] Carey, Studia Math. LXXXIII pp 19– (1986)
[4] Guivarc’h, Marches aléatoires sur les groupes de lie 624 (1977) · Zbl 0367.60081 · doi:10.1007/BFb0061339
[5] Dantzig, Compositio Math. 3 pp 408– (1936)
[6] DOI: 10.1007/BF01450491 · Zbl 0811.22004 · doi:10.1007/BF01450491
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