Let $G$ be a multiplicative group of modulo $16$; that is, $G=\{1,3,5,7,9,11,13,15\}$. Next, let $H=\{1,15\}$ and $K=\{1,9\}$. Determine whether $G/H$ and $G/K$ are isomorphic.
Attempt: Define a mapping $f:G/H \to G/K$ by $f(gH)=gK$ for all $g \in G$. This mapping is indeed a group homomorphism: Let $g_1H,g_2H \in gH$. Notice that $$f((g_1H)(g_2H)) = f(g_1g_2H) = g_1g_2K = (g_1K)(g_2K) = f(g_1H)f(g_2H).$$ But, I have a little bit trouble when showing injectivity. For any $g_1H,g_2H \in gH$ such that $f(g_1H)=f(g_2H)$, how to show that $g_1H=g_2H$?
Any ideas? Many thanks in advanced.
As said in comments, your way doesn't work.
Alternatively, notice that $\frac{G}{H}$ is cyclic group of order $4$ generated by $3H$, while $\frac{G}{K}$ is a non cyclic group of order $4$, as every element of $\frac{G}{K}$ had order $2$. Hence they are not isomorphic.