Affiliation:
1. School of Applied Mathematics , Nanjing University of Finance & Economics , Wenyuan Road No. 3, 210046 Nanjing , P. R. China
2. School of Computer Science , Reichman University , 8 Ha’universita Street , Herzliya 4610101 , Israel
Abstract
Abstract
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
/
N
G/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
)
c(G)
is compact and
G
/
c
(
G
)
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.
Subject
Applied Mathematics,General Mathematics