C-minimal topological groups

Wenfei Xi, Menachem Shlossberg

Research output: Contribution to journalArticlepeer-review


In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of [P. Hall and C. R. Kulatilaka, A property of locally finite groups, J. Lond. Math. Soc. 39 1964, 235-239] and a characterization of a certain class of Lie groups, due to [S. K. Grosser and W. N. Herfort, Abelian subgroups of topological groups, Trans. Amer. Math. Soc. 283 1984, 1, 211-223], we prove that a c-minimal locally solvable Lie group is compact. It is shown that a topological group G is c-(totally) minimal if and only if G has a compact normal subgroup N such that G/NG/N is c-(totally) minimal. Applying this result, we prove that a locally compact group G is c-totally minimal if and only if its connected component c(G) is compact and G/c(G) is c-totally minimal. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering [D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology. III, Atlantis Press, Paris 2014, 229-327, Question 3.10 (b)], we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.

Original languageEnglish
Pages (from-to)1641-1650
Number of pages10
JournalForum Mathematicum
Issue number6
StatePublished - 1 Nov 2021
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2021 Walter de Gruyter GmbH, Berlin/Boston.


The first author is supported by the National Natural Science Foundation of China (Grant No. 12001264)

FundersFunder number
National Natural Science Foundation of China12001264


    • C-minimal group
    • Hereditarily non-topologizable group
    • Lie group
    • Locally solvable group
    • Minimal group


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