Boolean powers: direct decomposition and isomorphism types

Abstract

We determine properties of Boolean powers of groups and other algebraic structures, and we generalize Jónsson’s theorem on Boolean powers of centerless, directly indecomposable groups. We show that every nonabelian, finitely generated group has 2 ℵ 0 {2^{{\aleph _0}}} nonisomorphic countable Boolean, and hence subcartesian, powers. We show that nonabelian groups G G such that either (i) G G is not the central product of two nonabelian groups or (ii) every pair of nontrivial normal subgroups of G G intersect nontrivially yield nonisomorphic Boolean powers with respect to nonisomorphic Boolean algebras.