IGNOU BMTE 144 2026 - Prove that every non-trivial subgroup of a cyclic group has finite ind

Question

Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that $(\mathbb{Q}, +)$ is not cyclic.

10 Jan 2026

Answer :

Word Count : 278

Let (G) be a cyclic group generated by (g), i.e., (G = \langle g \rangle), and let (H) be a non-trivial subgroup of (G). Since (G) is cyclic, every element of (H) can be written as a power of (g). Let (h) be a non-identity element of (H). Then (h = g^k) for some integer (k \neq 0). Consider the set of all integers (n) such that (g^n \in H). This set

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