**Problem**. Give an example of a non-abelian group that satisfies the following condition: for every and every in there exists a unique element such that

** Solution**. Consider and as additive groups and define the map by for all and See that is a group homomorphism. Let Then is non-abelian; in fact . To prove that has the required property, see that for any and integer we have So if and are given, then the only satisfying is where and

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