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.

Today we're resuming our informal chat on quotient groups.  Previously we said that belonging to a (normal, say) subgroup $N$ of a group $G$ just means you satisfy some property. For example, $5\mathbb{Z}\subset\mathbb{Z}$ means "You belong to $5\mathbb{Z}$ if and only if you're divisible by 5". And the process of "taking the quotient" is the simple observation that every element in $G$ either

#1) belongs to N            or            #2) doesn't belong to N

and noting that...