# 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 $G / H$.

$\blacksquare$