Cardinality and words in profinite groups

In this thesis, we study cardinalities and words in profinite groups. Let w = w(x1,...,xr) be a word, that is, an element of the free group on x1,...,xr. We are interested in the set of all the w-values in a group G, that is Gw = {w(g1,...,gr) : g1,...,gr ∈ G}, and the verbal subgroup w(G) generated by it. In particular we are interested in the sizes of these sets when w is a multilinear commutator word and G is a profinite group. A multilinear commutator word is a word obtained by nesting commutators and using each variable only once. For example the simple commutator [x, y] and more generally the higher commutators are multilinear commutator words. The goal of this thesis is to show that for a profinite group G and a multilinear commutator word w, if the cardinality of Gw is infinite then |Gw| = |w(G)| = 2a for some cardinal number a. The thesis is divided into three chapters. In the first one we will briefly introduce ordinal and cardinal numbers. In the second chapter we will show that the cardinal number of an infinite profinite group is 2a for some cardinal number a. In addition, we will prove that if a continuous map from a profinite group to a Hausdorff space has a "small image", then it is constant on a coset of a "big" closed subgroup. Finally in the third chapter we will collect some known results on multilinear commutator groups and we will prove our main results.