Welcome back to our little discussion on quotient groups! (If you're just now tuning in, be sure to check out "What's a Quotient Group, Really?" Part 1 and Part 2!) We're wrapping up this mini series by looking at a few examples. I'd like to take my time emphasizing intuition, so I've decided to give each example its own post. Today we'll take an intuitive look at the quotient given in the First Isomorphism Theorem.
Example #1: The First Isomorphism Theorem
But just how might an element $g\in G$ satisfy 2)? Well, $\phi(g)\neq e_H$, of course! But notice! There could be many elements besides $g$ who also map to the same $\phi(g)$ under $\phi$. (After all, we haven't required that $\phi$ be injective.) In fact, every element of the form $gg'$ where $g'\in\ker\phi$ fits the bill. So we group all those elements together in one pile, one coset, and denote it $g\ker\phi$. The notation for this is quite good: the little $g$ reminds us, "Hey, these are all the folks that map to the value of $\phi$ at that $g$." And multiplying $g$ by $\ker\phi$ on the right is suggestive of what we just observed: we can obtain other elements with the same image $\phi(g)$ by multiplying $g$ on the right by things in $\ker\phi$.
I like to imagine the elements of $G$ as starting off as dots scattered everywhere,
And this is why I like to think of $G/\ker\phi$ as "things in $G$ that don't map to the identity."
A closing remark
(By the way, there's nothing really special about groups here. There's a first isomorphism theorem for other algebraic objects, and the same intuition holds.)