The problem asks to determine the number of finite subgroups in the group $(\mathbb{Z}, +)$, where $\mathbb{Z}$ is the set of integers under addition.

Solution:

1. Key Property of Subgroups of $(\mathbb{Z}, +)$:

   • $(\mathbb{Z}, +)$ is an infinite cyclic group, and all its subgroups are of the form: $n\mathbb{Z} = \{nk : k \in \mathbb{Z}\}, \text{ for some integer } n \geq 0.$
   • Here, $n\mathbb{Z}$ denotes the set of all multiples of $n$.
   • Importantly, all subgroups of $(\mathbb{Z}, +)$ are infinite (except the trivial subgroup $\{0\}$, which is finite but contains only one element).

2. Finite Subgroups in $(\mathbb{Z}, +)$:

   • The only finite subgroup of $(\mathbb{Z}, +)$ is $\{0\}$, as no other subgroup can have a finite number of elements. This is because all $n\mathbb{Z}$ (for $n > 0$) are infinite.

3. Conclusion:

   • The number of finite subgroups in $(\mathbb{Z}, +)$ is exactly 1 (the trivial subgroup $\{0\}$).

Correct Answer: (b) 1

Would you like further clarification or examples on subgroups of $(\mathbb{Z}, +)$?

Here are 5 related questions to expand your understanding:

1. What are the infinite subgroups of $(\mathbb{Z}, +)$?
2. What is the structure of subgroups in a general cyclic group?
3. Can there exist any finite subgroup in the group of rational numbers $(\mathbb{Q}, +)$?
4. How can we prove that all subgroups of $(\mathbb{Z}, +)$ are cyclic?
5. What are the possible subgroups of the additive group $(\mathbb{Z}/n\mathbb{Z}, +)$ (integers modulo $n$)?

Tip: Always analyze the properties of the group operation (e.g., addition or multiplication) to identify possible subgroups.