Finding n-Power Extensions of Groups
AuthorTaylor, William Dixon
Mathematics and Statistics
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In this paper we consider the group-theoretic analogue of Galois Theory. That is, given a group G and a natural number n, we find groups H such that G can be embedded into H in such a way that every element of G has an nth root in H. It is not difficult to see (and we prove) that given any group G and natural number n such an extension of G exists. In this paper, we attempt to find the smallest order of such an extension, which we will call the minimal n-index of G. We answer the question completely for cyclic groups. Further, we examine certain ways of constructing new groups from old, in particular the direct and semidirect product, and determine how these constructions interact with the minimal n-index of G. We conclude with some conjectures regarding the minimal n-index and some questions to inspire further research.