Okay so I have this question:
Let $G$ be a group and suppose that $N$ and $K$ are normal subgroups
of $G$, where $N \leq K$. Define a map: $\theta:G/N \rightarrow G/K$
by $\theta(aN)=aK$. Show that $\theta$ is well defined.
I know that in order to show a map is well defined, I have to show that $x_1=x_2 \implies \theta(x_1)=\theta(x_2)$. So for this question, is it a case of proving $aN_1 = aN_2 \implies aK_1=aK_2$? I've proved maps are well defined in the past, but I just don't understand the question here, can someone explain what it is I have to prove please? Thanks in advance.
Best Answer
That's almost it. What you actually want to do is show that $a_1N=a_2N$ implies that $a_1K=a_2K$, since the inputs into $\theta$ are different left cosets of $N$.