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) ⊇ … ⊇ Ωn(G) …
of closed characteristic subgroups of G with two important properties, namely, that their intersection
∩∞n=1 Ωn(G) is
identity component of the center of the identity component G0 of G, and that each
quotient group Ωn−1(G)/Ωn(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 , 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.