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Master's Thesis from the year 2013 in the subject Mathematics - Algebra, grade: -, University of Warwick (Institute of Mathematics), course: M.Sc dissertation in Pure Mathematics, language: English, abstract: We start with a brief introduction to Free Groups, thereby appreciating Nielsen’s approach to the Subgroup theorem. Beautiful results of J. H. C. Whitehead, J. Nielsen, E. S. Rapaport, Higgins and Lyndon, and J. McCool form our building block. We study different automorphisms of a finitely generated free group as well as a finite set of automorphisms which Whitehead used to deduce that if two elements of a finitely generated free group are equivalent under an automorphism of the group, then they are equivalent under such automorphisms. We write program aimed at appreciating Whitehead’s theorem, starting with programs for appreciating Whitehead automorphisms to programs for determining whether two elements of a finitely generated free group are equivalent or not. We conclude by classifying all minimal words of lengths 2, 3, 4, 5 and 6 in F n (for some n ∈ [2, 6]) up to equivalence.