An example of a Q-minimal precompact topological group containing a nonminimal closed normal subgroup

Abstract

In 1977, I. Prodanov [~ showed that closed subgroups of abellen (q-)mlnlmal topological groups are again (q-) mlnlmsl. These permanence properties do not carry over to the nonabellen case. In fact, in [3J a q-mlnimal topological group X was constructed, containing a nonminlmal closed subgroup G. However, in that example X is not a SIN-group end G is not e normal subgroup of X.In this note we give an example of a q-minimal topological group which is precompact (hence a SIN-group), mstrizeble, but contains a nonminimal closed normal subgroup. This example also gives a negative answer to a problem posed by D.L.

Quotients of locally minimal groups

“…Minimality fails to be preserved under taking quotients. This is why the smaller class of totally minimal groups, namely the minimal groups that are minimal along with all their Hausdorff quotients, was introduced in [16] (somewhat later also in [36], under the name q-minimal groups). Equivalently, a topological group G is totally minimal if every surjective continuous homomorphism of G onto a Hausdorff topological group is open.…”

“…The following notion was proposed in [16] (somewhat later also in [36]): a subgroup H of a topological group G is called totally dense if H ∩ N is dense in N for every closed normal subgroup N of G. This notion was used to provide the following crucial criterion for total minimality of dense subgroups:…”

“…Obviously, G is minimal precisely when G satisfies the open mapping theorem with respect to continuous isomorphisms with domain G. Compact groups are minimal, the first examples of non-compact minimal groups were found by Doïchinov [23] and Stephenson [37]. The research in this field was inspired by a challenging problem set by G. Choquet at the ICM in Nice 1970; it was quite intensive for almost five decades (see [4,10,[16][17][18][19]9,24,29,31,[33][34][35][36]38], as well as the surveys or monographs [5,6,11,12,14,22]).…”

Every topological group is a group retract of a minimal group

“…A Hausdorff topological group G is minimal (introduced by Stephenson [33] and Doïchinov [9]) if G does not admit a strictly coarser Hausdorff group topology. Totally minimal groups are defined by Dikranjan and Prodanov [7] as those Hausdorff groups G such that all Hausdorff quotients are minimal (later these groups were studied also by Schwanengel [31] under the name q-minimal groups). First we recall some facts about minimality; mainly concerning the purposes of the present work.…”