The commutator subgroup G' of a group G is the subgroup generated by the set {x–1y–1xy | x, y ∈ G}. (That is, every element of G' has the form a1 i1a2 i2 … ak ik, where each aj has the form x–1y–1xy, each ij = ±1, and k is any positive integer.) Prove that G' is a characteristic subgroup of G. (This subgroup was first introduced by G. A. Miller in 1898.)
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