I apologize for the long post, but I'm currently a student finishing up his first semester in group theory. My introduction was pretty definition-heavy so I've found I can internalize concepts (such as quotient groups, normal subgroups, etc.) myself by forming my own way of motivating and teaching them intuitively. I'd like to know if my current presentation/understanding is correct.

I think after learning about subgroups and Lagrange's theorem, a natural question is then if we can break down a group $G$ to better understand its parts and hopefully the whole $G$ (as is tradition in any analytical endeavor). But if we want to pull back any useful understanding of $G$ from this smaller group, it ought to preserve some structure of $G$. That structure is exactly how the operation acts on elements of $G$ (since groups are just elements with an operation relating them).

So for the sake of exploration, we pretend to have the magic function $\phi : G \rightarrow H$ that does exactly this for us–maps $(G, *_G)$ to some smaller part $(H, *_H)$, then ask what we can say about $\phi$. Our original goal was for $\phi$ to preserve the operation, i.e. for all $a, b \in G$ that $\phi(a *_G b) = \phi(a) *_H \phi(b)$.

The next thing I would observe is that since $H$ has smaller order, $\phi$ necessarily maps a multiple elements, say $a, b$, in $G$ to the same element in $H$. In this sense, $a$ and $b$ are "equivalent" under $\phi$. Given that we have "willed" $\phi$ to operation-preserving, we can see that a natural way this arises by letting $ak = b$ for some $k \in G$:

$$ak = b \implies \phi(b) = \phi(a *_G k) = \phi(a) *_H \phi(k).$$

If we want $\phi(b) = \phi(a) *_H \phi(k) = \phi(a)$ then $\phi(k) = e_H$. I think leads naturally to the definition of the kernel: it's a set of elements that maps to the identity, and makes $a$ equivalent to $b$ mod $\ker \phi$. And in fact, as I've learned from this answer, we naturally get equivalence classes of elements that partition the group into cosets analogous to modular arithmetic. So, $\phi$ takes elements and puts them neatly into these equivalence classes (abstracting away some of the details in $G$ that look "the same" in $H$, leading–at least for me–directly to the First Isomorphism Theorem). Then it makes sense to propose the map $\phi : g \mapsto g \ker \phi$.

The next question becomes what the operation of this $H$ looks like. We've established that elements of $H$ are cosets (and equivalence classes), so for two elements in cosets $g_1 \ker\phi$ and $g_2 \ker\phi$, once combined by $*_H$ we'd want for the result to be in $g_1g_2 \ker\phi$ (modular arithmetic analogy works here as well). Set-wise, we might write

$$g_1\ker\phi \cdot g_2\ker\phi = g_1g_2\ker\phi.$$

But does this come for free? For an element $g_1k_1g_2k_2 \in g_1\ker\phi \cdot g_2\ker\phi$ to look like $g_1g_2k$ for some $k$, it must be that $k_1g_2 = g_2k_3$ for some $k_3$. Set-wise this can be written as $g\ker\phi = \ker\phi g$, i.e. left-costs = right-cosets and it turns out it indeed satisfies this condition and we are safe to proceed.

So, in the end, we have designed $H$, a broken-down version of $G$. And how did we do it? By "dividing out" or "quotienting out" the information that looks the same under $\phi$ in $H$–$\ker \phi$. Thus we write $H$ as $G/\ker\phi$, aptly called a quotient group.

Although, you could flip this presentation, and instead of viewing from the kernel perspective, suppose $K$ is some arbitrary group. Then it must satisfy the condition of left-cosets = right-cosets (which we name normality because it is a nontrivial property that gives us a usable quotient) for $G/K$ to be a group, as $\ker\phi$ already does, and through satisfying normality automatically becomes the kernel of some homomorphism (namely the natural, which I've presented).

My questions are:

• Is this presentation correct (on an intuitive level, I know there are lots of places for concrete proofs)? It feels right to me, but I also feel like I may have gotten definitions vs. implications mixed up.
• If so, does any textbook follows this approach that I can dig into?
• I think homomorphisms also can fit into this framework, given I suggest $\phi$ pretty early on, but how would non-surjective homomorphisms be explained?