A profinite group is a compact Hausdorff topological group whose open subgroups form a neighborhood base of the identity. We define the Shalev property for finitely generated profinite groups, which essentially means that a bound can be placed on the number of n-th powers needed to express an element of any n-th power subgroup. In a finitely generated profinite group with the Shalev property, every subgroup of finite index is open and thus, the topological structure is completely determined by the algebraic structure. The aim of this dissertation is to show that every profinite group of finite rank has the Shalev property. We show that in a finitely generated profinite group with the Shalev property, each subgroup of finite index also has the Shalev property. We show further that, for finitely generated profinite groups, the Shalev property is extension closed. We then use this result to show that every soluble profinite groups, of finite rank has the Shalev property. Using this fact, together with the structure theorem for profinite groups of finite rank, which states that these groups are pro-nilpotent-by-soluble-by-finite, we show that every profinite group of finite rank has the Shalev property.
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