We prove a new structure theorem which we call the Countable Layer Theorem. It says that for any compact group
Gwe can construct a countable descending sequence G= Ω0( G) ⊇ … ⊇ Ω n( G) … of closed characteristic subgroups of Gwith two important properties, namely, that their intersection ∩∞ n=1 Ω n( G) is Z0( G0), the identity component of the center of the identity component G0 of G, and that each quotient group Ω n−1( G)/Ω 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).
In the special case that
Gis totally disconnected (that is, profinite) the intersection of the sequence is trivial. Thus, even in the case that Gis profinite, our theorem sharpens a theorem of Varopoulos [ 8], who showed in 1964 that each profinite group contains a descending sequence of closed subgroups, each normal in the preceding one, such that each quotient group is a product of finite simple groups. Our construction is functorial in a sense we will make clear in Section 1.