Question

Let G be a group. Let G′ be the collection of all products in G of the form a₁^i₁a₂^i₂ ···a_k^i_k where each a_j itself has the form x^−1y^−1xy, each i_j = ±1, and k is any positive integer.

(a) Show that G′ is a subgroup of G. Conclude that G′ is the subgroup of G generated by the set S = { x^−1y^−1xy | x,y ∈ G }.

(b) Prove that G′ is normal in G.

(c) Prove that the factor group G/G′ is Abelian.

(d) If N is a normal subgroup of G and G/N is Abelian, prove that G′ is a subgroup of N.

(e) Prove that if H is a subgroup of G and G′is a subgroup of H , then H is normal in G.