3. Let G be a group of odd order and H be a normal subgroup. (a) Prove that if |H| = 5, then H ≤ Z(G), where Z(G) is the center of G. (b) Prove the existence of a non-abelian group of order 21. (Hint: You might try calculating the normalizer and centralizer of the 7-cycle (1234567) in S7. Or look for a subgroup of the 2 × 2 invertible matrices over Z7.)

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