9.3 Nilpotent Groups
1. a. Z(A4) = {ε} and Z(An) = {ε} if n ≥ 5 because An is simple and nonabelian. Proceed as in Example 1.
2.
1. [H, K] = [K, H] because [h, k]−1 = [k, h], so inverses of products of commutators are again products of commutators.
3. If H 
G and K G then a−1[h, k]a = [a−1ha, a−1ka] 
[H, K] for all h H, k K. Hence if c1, c2, . . ., ck are commutators, then
G and K G then a−1[h, k]a = [a−1ha, a−1ka] [H, K] for all h H, k K. Hence if c1, c2, . . ., ck are commutators, then
is a product of commutators.
3. α([h, k]) = [α(h), α(k)] for all h H, k K, so α([H, K]) ⊆ [α(H), α(K)]. Since each element of [α(H), α(K)] is a product of commutators of the form [α(h), α(k)] = α[h, k], this is equality. The rest is clear if α : G → G is any inner automorphism (any automorphism).
H, k K, so α([H, K]) ⊆ [α(H), α(K)]. Since each element of [α(H), α(K)] is a product of commutators of the form [α(h), α(k)] = α[h, k], this is equality. The rest is clear if α : G → G is any inner automorphism (any automorphism).
5. One verifies ...
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