Applications of the Isomorphism theorems

In my study of groups, rings, modules etc, I've seen the three isomorphism theorems stated and proved many times. I use the first one ( $G/\ker \phi \cong \operatorname{im} \phi$ ) very often, but I can't recall having ever used the other two. Can anyone give some examples where they are used in a crucial way in some proof?

For clarity, let us say that the 2nd one is : $(M/L)/(N/L) \cong M/N$ under the appropriate conditions, and the 3rd one is $(M+N)/N \cong M/(M\cap N).$

This is an application of the second isomorphism theorem, although the theorem does not play a crucial role in it.

Let $a, b$ be positive, say, integers. Then $$ a \mathbf{Z} + b \mathbf{Z} = \gcd(a, b) \mathbf{Z}, $$ and $$ a \mathbf{Z} \cap b \mathbf{Z} = \operatorname{lcm}(a, b) \mathbf{Z}. $$ Now the second isomorphism theorem gives you the isomorphism $$ \frac{\gcd(a, b) \mathbf{Z}}{b \mathbf{Z}} = \frac{a \mathbf{Z} + b \mathbf{Z}}{b \mathbf{Z}} \cong \frac{a \mathbf{Z}}{a \mathbf{Z} \cap b \mathbf{Z}} = \frac{a \mathbf{Z}}{\operatorname{lcm}(a, b) \mathbf{Z}}. $$ Comparing orders you get $$ \frac{b}{\gcd(a, b)} = \frac{\operatorname{lcm}(a, b)}{a}, $$ which is the well-known formula $$ \gcd(a, b) \operatorname{lcm}(a, b) = a b. \tag{gcd/lcm} $$ Clearly (gcd/lcm) can be proved without recourse to the second isomorphism theorem. But whenever I teach the theorem, I find it useful for the students to show them that we are, in a sense, generalizing a fact they are already familiar with.

Here and here are examples of questions I've answered with little more than the second isomorphism theorem.

The second one comes up a lot in modular arithmetic.

If you working in the integers modulo $16$? Maybe you'll want to reduce modulo $8$. Can you do that? And if so, do you get the integers modulo $8$?

$$ (\mathbb{Z} / 16\mathbb{Z}) / (8\mathbb{Z} / 16\mathbb{Z}) \cong \mathbb{Z} / 8\mathbb{Z} $$

Ah, phew!

The second isomorphism theorem often comes up when you want to do calculations with a quotient ring by lifting the problem to the original ring (e.g. because it is easier to work with, such as the integers or being a polynomial ring)

Now I'm working in the even numbers but I need to also work modulo $3$. How can I make sense of that? I suppose I need to mod out by the intersection of $3\mathbb{Z} \cap 2\mathbb{Z}$. What does that work out to?

$$2\mathbb{Z} / (2\mathbb{Z} \cap 3 \mathbb{Z}) \cong (2 \mathbb{Z} + 3 \mathbb{Z}) / 3 \mathbb{Z} \cong \mathbb{Z} / 3\mathbb{Z}$$

The third isomorphism theorem often comes up when you have several moduli to work with, and want to understand one ideal modulo the other.

As Boris and Alexander have mentioned, both these theorems are used a lot in the study of normal series, and particularly soluble groups.

One basic example of this is the following. A group $G$ is metabelian if it has a normal subgroup $N$ such that $N$ and $G/N$ are both abelian.

Theorem Any subgroup of a metabelian group is metabelian.

Proof Let $G$ be a metabelian group with normal abelian subgroup $N$ such that $G/N$ is abelian. Let $H$ be any subgroup of $G$. Then $H\cap N$ is a normal abelian subgroup of $H$ and $H/(H\cap N)\cong HN/N$, which is a subgroup of $G/N$ and is hence abelian. Thus $H$ is metabelian. $\Box$

The same sort of idea could be used with induction to prove that any subgroup of a soluble group of derived length $n$ is also soluble, of derived length at most $n$, although I don't think that's the standard proof.

Both are used a lot when studying normal series, for example derived, Fitting, lower/upper central, or composition series, and explicitly talking about its members.

If I'm trying to prove something about a Sylow $p$-subgroup of a quotient by a Sylow $q$-subgroup $(p\not= q)$, for example, $PQ/Q\cong P/(P\cap Q)$ is very useful because we know that $P\cap Q=1$. So we have $PQ/Q\cong P$, a intuitive result that the internal a Sylow subgroup is not affected by quotienting out by another unrelated Sylow subgroup. Plus, then in the rest of the proof, we know we can refer to the Sylow $p$-subgroups of $G/Q$ as $P$ instead of $PQ/Q$, which is notationally convenient.

Similarly, say we're looking at the series $Z_0=G$ and $Z_i/Z_{i-1}=Z(G/Z_{i-1})$. We know that $\frac{G/Z_{i-1}}{Z_i/Z_{i-1}}\cong G/Z_i$, which is a much easier way of talking about that group.