Denote $R=\langle6\rangle$ and $S=\langle30\rangle$.
Consider the subgroups $R$ and $S$ 0f the set of integers $\mathbb{Z}$.
Show that $R/S \cong \mathbb{Z}_5$.
$(R/S$ is the set of left cosets of $S$ over $R$, $\mathbb{Z}_5$ denotes congruence class $\pmod 5.)$
Define a map $f:R/S \to \mathbb{Z}_5$ by $f(6m+30\mathbb{Z})=(m+5\mathbb{Z})$.
I have shown the following:
- $f$ is well-defined.
- $f$ is a homomorphism.
- $f$ is surjective or onto.
Now, to show that $f$ is injective, I assume that $f(6m+30\mathbb{Z})\equiv f(6n+30\mathbb{Z}) \pmod 5$ My question is: will it imply that $m \equiv n \pmod 5$?
If yes, please show me how.