Subgroup of Cyclic Group is Cyclic/Proof 3

Theorem

Let $G$ be a cyclic group.

Let $H$ be a subgroup of $G$.

Then $H$ is cyclic.

Proof

Let $G$ be a cyclic group generated by $a$.

Let $H$ be a subgroup of $G$.

By Cyclic Group is Abelian, $G$ is abelian.

By Subgroup of Abelian Group is Normal, $H$ is normal in $G$.

Let $G / H$ be the quotient group of $G$ by $H$.

Let $q_H: G \to G / H$ be the quotient epimorphism from $G$ to $G / H$:

$\forall x \in G: \map {q_H} x = x H$

Then from Quotient Group Epimorphism is Epimorphism, $H$ is the kernel of $q_H$.

From Kernel of Homomorphism on Cyclic Group, $H$ is a cyclic group.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $7$: Homomorphisms: Exercise $9$