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Let $Z$ be the set of all integers. Let $n \in Z$ and $nZ = \{nk : k \in Z \}.$ We know that $Z$ is a group under addition operation.
Which of the following is/are true?

  1. $nZ$ is a subgroup of $Z$ (under addition operation) for all $n \in Z.$
  2. Every subgroup of $(Z,+)$ is isomorphic to $nZ$ for some $n.$
  3. $3Z$ is the smallest subgroup of $(Z,+)$ containing $3.$
  4. $nZ$ is a cyclic subgroup of $Z.$
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All the statements are true. We have all these in our lectures.
$(Z,+)$ is a cyclic group, and every subgroup of a cyclic group is cyclic.
Answer:

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