Let G be a group and let G' be the subgroup of G generated
Chapter 9, Problem 62E(choose chapter or problem)
Let \(G\) be a group and let \(G\)' be the subgroup of \(G\) generated by the set \(S=\left\{x^{-1} y^{-1} x y \mid x, y \in G\right\}\). (See Exercise 3, Supplementary Exercises for Chapters 5–8, for a more complete description of \(G\)'.)
a. Prove that \(G\)' is normal in G.
b. Prove that \(G/\)\(G\)' is Abelian.
c. If \(G/N\) is Abelian, prove that \(G\)' \(\leq\) \(N\).
d. Prove that if \(H\) is a subgroup of \(G\) and \(G\)' \(\leq\) \(H\), then \(H\) is normal in \(G\).
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