Key subgroups in topological groups

Abstract: We introduce two minimality properties of subgroups in topological groups. A subgroup $H$ is a key subgroup (co-key subgroup) of a topological group $G$ if there is no strictly coarser Hausdorff group topology on $G$ which induces on $H$ (resp., on the coset space $G/H$) the original topology. Every co-minimal subgroup is a key subgroup while the converse is not true. Every locally compact co-compact subgroup is a key subgroup (but not always co-minimal). Any relatively minimal subgroup is a co-key subgroup (but not vice versa). Extending some results concerning the generalized Heisenberg groups, we prove that the center ("corner" subgroup) of the upper unitriangular group $\mathrm{UT(n,K)}$, defined over a commutative topological unital ring $K$, is a key subgroup. Every "non-corner" 1-parameter subgroup $H$ of $\mathrm{UT(n,K)}$ is a co-key subgroup. We study injectivity property of the restriction map $$r_H \colon \mathcal{T}{\downarrow}(G) \to \mathcal{T}{\downarrow}(H), \ \sigma \mapsto \sigma|H$$ and show that it is an isomorphism of sup-semilattices for every central co-minimal subgroup $H$, where $\mathcal{T}{\downarrow}(G)$ is the semilattice of coarser Hausdorff group topologies on $G$.