Regular Homeomorphisms of Finite Order on Countable Spaces

We present a structure theorem for a broad class of homeomorphisms of finite order on countable zero dimensional spaces. As applications we show the following. (a) Every countable nondiscrete topological group not containing an open Boolean subgroup can be partitioned into infinitely many dense subsets. (b) If G is a countably infinite Abelian group with finitely many elements of order 2 and beta G is the Stone-Cech compactification of G as a discrete semigroup, then for every idempotent p is an element of beta G \ {0}, the subset {p, -p} subset of beta G generates algebraically the free product of one-element semigroups {p) and {-p}.