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Let Z be the integers. (a) Let C1 = {(a, a) | a ∈ Z}. Prove...

Let Z be the integers.

(a) Let C1 = {(a, a) | a ∈ Z}. Prove that C1 is a subgroup of Z × Z.

(b) Let n ≥ 2 be an integer, and let Cn = {(a, b) | a ≡ b( mod n)}. Prove that Cn is a subgroup of Z × Z.

(c) Prove that every proper subgroup of Z × Z that contains C1 has the form Cn for some positive integer n.

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