Let n > 1 be a fixed integer and let G be a group. If the set H = {x∈G | |x| = n} together with the identity forms a subgroup of G, prove that it is a normal subgroup of G. In the case where such a subgroup exists, what can be said about n? Give an example of a non-Abelian group that has such a subgroup. Give an example of a group G and a prime n for which the set H together with the identity is not a subgroup.
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