C-minimal topological groups

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