TITLE: About how to understand the third isomorphism theorem
QUESTION [5 upvotes]: What is a good way to understand the intuition behind the third isomorphism theorem? Is it something looking like zooming out of the group structure?(i.e. discard the detailed info by modding out a normal subgroup).

REPLY [2 votes]: Your suggestion is fine.
Another intuition I think is important is that it's something to understand in the context of higher algebra: you're doing arithmetic with groups. Not with elements of groups, but with the groups themselves. The third isomorphism theorem is a basic algebraic identity used either for simplifying quotients of quotients, or conversely to write a quotient as a quotient of quotients as needed.
I often use it in the context of expressing a quotient $G/(HK)$ as first modding out by $H$ and then modding out by $K$, which can be justified as
$$ G / (HK) \cong (G / H) / (HK/H) \cong (G/H) / \pi(K) $$
where $\pi$ is the projection map $G \to G/H$.
I find the third isomorphism particularly appealing if you think in terms of setoids -- that is, you think in terms of sets equipped with equivalence relations, and modding out by a normal subgroup means equipping your group with the corresponding equivalence relation rather than constructing the set of equivalence classes. The third isomorphism theorem then says if you equip things with an equivalence relation, and then a coarser equivalence relation, you get the same result as if you just skipped the intermediate step.