IGNOU BMTC 134 2026 - Prove that every non-trivial subgroup of a cyclic group has fini

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 : 202

Let (G = \langle g \rangle) be a cyclic group and let (H) be a non-trivial subgroup of (G). Since (G) is cyclic, every element of (G) can be written as (g^n) for some integer (n). Let (h \in H) be a non-identity element. Then (h = g^m) for some nonzero integer (m). The subgroup (H) is itself cyclic, generated by (g^m), so (H =

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