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- Prove that if r and s are relatively prime positive integers, then any cyclic group of order rs is the direct sum of a cyclic group of order r and a cyclic group of order s.Show that every subgroup of an abelian group is normal.Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.
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- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.24. Let be a cyclic group. Prove that for every normal subgroup of , is a cyclic group.