A structure theorem on compact groups (Q2724998)

The authors prove a new structure theorem which they call countable layer theorem. It says that for any compact group \(G\) we can construct a countable descending sequence \(G=\Omega_0(G)\supseteq\dots \supseteq\Omega_n(G)\supseteq\dots\) of closed characteristic subgroups of \(G\) with two important properties, namely, that their intersection \(\bigcap_{n=1}^\infty\Omega_n(G)\) is \(Z_0(G_0)\), the identity component of the center of the identity component \(G_0\) of \(G\), and that each quotient group \(\Omega_{n-1}\setminus\Omega_n(G)\), is a cartesian product of compact simple groups (that is, compact groups having no normal subgroups other than the singleton and the whole group).