Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that  is not cyclic.

To prove that every non-trivial subgroup of a cyclic group has finite index, let's first establish some definitions and properties:

1. **Cyclic Group:** A group \( G \) is cyclic if there exists an element \( a \) in \( G \) such that every element of \( G \) can be expressed as \( a^n \) for some integer \( n \).
  
2. **Index of a Subgroup:** If \( H \) is a subgroup of a group \( G \), the index of \( H \) in \( ____ ______ ___ ______ _______ ____.
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