Commutativity conditions for DMAP solvable topological groups (Q2918437)

For any topological group \(G\), a closed normal subgroup (and even a characteristic subgroup) of \(G\), formed by all elements \(g\in G\) such that \(\phi(g)=1\) for every almost periodic function \(\phi\) on \(G\) is introduced; if this subgroup, obviously coinciding with the intersection of kernels of all finite-dimensional continuous complex unitary representations of \(G\) (or equivalently, with the intersection of the kernels of all irreducible finite dimensional continuous complex unitary representations of \(G\)), is the identity subgroup for a topological group \(G\), then \(G\) is said to be maximally almost periodic (MAP).NEWLINENEWLINEA DMAP topological group means a topological group which has sufficiently many (not necessarily continuous) finite-dimensional representations, i.e., a topological group which is an MAP group when equipped with the discrete topology.NEWLINENEWLINEThe basic result of the paper is as follows. A solvable DMAP topological group having a chief series all of whose Abelian factors are divisible, is commutative.