Minimality of the inner automorphism group

By [7], a minimal group G is called z-minimal if G/Z(G) is minimal. In this paper, we present the z-Minimality Criterion for dense subgroups. For a locally compact group G, let Inn(G) be the group of all inner automorphisms of G, endowed with the Birkhoff topology. Using a theorem by Goto [15], we obtain our main result which asserts that if G is a connected Lie group and H∈{G/Z(G),Inn(G)}, then H is minimal if and only if H is centre-free and topologically isomorphic to Inn(G/Z(G)). In particular, if G is a connected Lie group with discrete centre, then Inn(G) is minimal. We prove that a connected locally compact nilpotent group is z-minimal if and only if it is compact abelian. In contrast, we show that there exists a connected metabelian z-minimal Lie group that is neither compact nor abelian. As in the papers [27,33], some applications to Number Theory are provided.