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I am reading again about free vector spaces over a set. In the Theory of Groups by Kurosh part of the construction of a free group is to construct a set of elements inverse to the those of the original set which can effectively "cancel" each other out if juxtaposed in a word made from elements of these two sets. I understand the mechanics of the constuction and the reduction of words but as a first question i would like to ask what is the inverse of an element of a set, if such a thing exists, apart from having this defined "cancellation" property

Matheinste.

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# Free groups.

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