Let Z be the integers. (a) Let C1 = {(a, a) | a € Z}. Prove that C1 is a subgroup of Z x Z. (b) Let n > 2 be an integer, and let Cn = {(a,b) | a = b( mod n)}. Prove that C, is a subgroup of Z × Z. : Prove that every proper subgroup of Z × Z that contains C1 has the (c) form Cn for some positive integer n.

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