Key subgroups in topological groups

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 from [M. Megrelishvili, Group representations and construction of minimal topological groups, Topology Appl. 62 (1995), no. 1, 1–19] and [D. Dikranjan and M. Megrelishvili, Relative minimality and co-minimality of subgroups in topological groups, Topology Appl. 157 (2010), no. 1, 62–76] concerning the generalized Heisenberg groups, we prove that the center (“corner” subgroup) of the upper unitriangular group UT(n, K), defined over a commutative topological unital ring K, is a key subgroup. Every “non-corner” 1-parameter subgroup H of UT(n, K) is a co-key subgroup. We study injectivity property of the restriction map r_H : T_↓(G) → T_↓(H), σ →→ σ|_H and show that it is an isomorphism of sup-semilattices for every central co-minimal subgroup H, where T_↓(G) is the semilattice of coarser Hausdorff group topologies on G.