Quotient Group of Cyclic Group/Proof 1
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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 $G$ be a cyclic group generated by $g$.
Let $H \le G$.
We need to show that every element of $G / H$ is of the form $\left({g H}\right)^k$ for some $k \in \Z$.
Suppose $x H \in G / H$.
Then, since $G$ is generated by $g$, $x = g^k$ for some $k \in \Z$.
But $\left({g H}\right)^k = \left({g^k}\right) H = x H$.
So $g H$ generates $G / H$.
$\blacksquare$