A structure theorem on compact groups

Abstract

We prove a new structure theorem which we call the Countable Layer Theorem. It says that for any compact group G we can construct a countable descending sequence G = Ω₀(G) ⊇ … ⊇ Ω_n(G) … of closed characteristic subgroups of G with 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 G is totally disconnected (that is, profinite) the intersection of the sequence is trivial. Thus, even in the case that G is 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.