Let G1 and G2 be groups, and let G = G1 × G2. Write e and e2 for the identities of G1 and G2 respectively. Let H = {(g1, e2) | gi € G1} and K = {(e1, g92) | g2 E G2}. (a) Prove that H and K are subgroups of G. (b) Prove that HK = KH =G. (c) Prove that HN K = {(e1, e2)}.

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