Minimality of the inner automorphism group

By [7], a minimal group G is called z-minimal if $G/Z\left(G\right)$ is minimal. In this paper, we present the z-Minimality Criterion for dense subgroups. For a locally compact group G, let $\mathrm{Inn}\left(G\right)$ 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 $H\in \{G/Z(G),\mathrm{Inn}(G\left)\right\}$, then H is minimal if and only if H is centre-free and topologically isomorphic to $\mathrm{Inn}(G/Z(G\left)\right)$. In particular, if G is a connected Lie group with discrete centre, then $\mathrm{Inn}\left(G\right)$ 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.