Compact Lie groups and factor spaces with respect to tori (Q1337847)

A toral subgroup \(T\) of a compact connected Lie group \(G\) is called irreducibly imbedded if \(T\) is a maximal torus of the commutator subgroup of the centralizer of a torus in \(G\), and if \(T\) cannot be constructed in the same way by means of a proper simple subgroup of \(G\). Let \(K = \prod^ s_{i = 1} K_ i\), where \(K_ i\) is a simply connected simple compact Lie group of rank \(> 1\) for all \(i\), and \(U = \prod^ s_{i = 1} U_ i\), where \(U_ i\) is an irreducibly imbedded toral subgroup of \(K_ i\). The main result of the paper is that any transitive action of a simply connected compact Lie group, having \(\leq s\) simple factors, on \(K/U\) is similar to the standard action of \(K\).