I'm reading the section on quotient groups in Dummit and Foote, and they give somewhat non-standard definition of a quotient group. I was wondering whether there is an easy way to see right away for someone who is familiar with the standard definition of a quotient group that DF's definition is equivalent to the standard one?
What I can see. The standard definition defines $G/K$ (as a set) as the set of left cosets of $K$ in $G$. Every such coset is an equivalence class under the equivalence relation on $G$ given by $g_1\sim g_2\iff g_1=g_2k$ for some $k\in K$. The fibers of $\phi:G\to H$ are the equivalence classes on $G$ given by $g_1\sim g_2\iff \phi(g_1)=\phi(g_2)$. So there are two partitions of $G$, and I think I need to see (1) why they are the same, (2) why multiplication is "the same".
Note that if $g_1 = g_2k$, then $\varphi(g_1) = \varphi(g_2)$ since $\varphi(k) = e$. Thus elements in the same coset are in the same fiber of $\varphi$.
If $g_1$ and $g_2$ are in the same fiber of $\varphi$, then $\varphi(g_1) = \varphi(g_2)$, so that $g_1g_2^{-1}\in K$, or $g_1 = g_2k$ for some $k\in K$.