Quotient Group of Cyclic Group/Proof 2
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Theorem
Let $G$ be a cyclic group which is generated by $g$.
Let $H$ be a subgroup of $G$.
Then $g H$ generates $G / H$.
Proof
Let $H$ be a subgroup of the cyclic group $G = \gen g$.
Then by Homomorphism of Powers for Integers:
- $\forall n \in \Z: \map {q_H} {g^n} = \paren {\map {q_H} g}^n = \paren {g H}^n$
As $G = \set {g^n: n \in \Z}$, we conclude that:
- $G / H = q_H \sqbrk G = \set {\paren {g H}^n: n \in \Z}$
Thus, by Epimorphism from Integers to Cyclic Group, $g H$ generates $G / H$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Theorem $25.3$