Section 1.4 A fundamental isomorphism theorem for groups, rings, vector spaces
IfTheorem 1.4.1. Fundamental theorem for group homomorphisms.
Let
Moreover, the image of
Proof.
The condition that \(\varphi(g) = \varphi_*(\pi(g))\) requires that
from which it is immediate that there is only one possible definition for \(\varphi_*\) (hence uniquely defined), and also that the images of \(\varphi\) and \(\varphi_*\) are the same.
Presuming the map \(\varphi_*\) is well-defined, we see that it is a group homomorphism since
where we have used the group operation on \(G/K\) and that \(\varphi\) is a group homomorphism.
The most important issue is that the map makes sense, i.e., is well-defined. In this case, we must check that
But
since \(\varphi(k) = e\) because \(K \subseteq \ker\varphi.\)
Finally, we compute
but that says
Theorem 1.4.3. First Isomorphism Theorem.
Let
Proof.
Let \(K=\ker\varphi\text{.}\) The fundamental theorem gives us a surjective homomorphism \(\varphi_*: G/K \to \Im \varphi\) whose kernel is \(K/K = eK = e_{G/K}\text{,}\) the identity of \(G/K\text{,}\) so the map \(\varphi_*\) is injective.
Remark 1.4.4.
The importance of the fundamental theorem
cannot be overstated. The main takeaway is that if ever
faced with the job of finding a homomorphism
Example 1.4.5. The canonical example?
For a positive integer
Of course we could write down a map taking \(m+n\Z \mapsto [m]_n\text{,}\) but then we would have to check that it is well-defined, a homomorphism, and eventually an isomorphism.
Instead we invoke the fundamental theorem. There is certainly a natural map \(\varphi:\Z\to \Z_n\) which takes an integer \(m\) to its residue class \([m]_n\) modulo \(n.\) It is a homomorphism by virtue that \(\Z_n\) is a group:
and it is certainly surjective. What is the kernel of \(\varphi\text{?}\)
By the first isomorphism theorem, the result is achieved.
Theorem 1.4.6. A meta fundamental theorem for homomorphisms.
Let
Proof.
Because all these structures have underlying group structures, that map \(\varphi_*\) is uniquely determined, well-defined and has the correct kernel and image. The only thing missing is to verify that \(\varphi_*\) has the additional properties necessary to be a homomorphism of the correct type (e.g., ring or vector space). But this is easily checked. For example, if the objects were rings, then \(Z\) would necessarily be an ideal. The map \(\varphi_*\) takes \(x+Z\) to \(\varphi(x)\text{.}\) We check (using the ring structure of \(X/Z\) and that \(\varphi\) is a ring homomorphism) that
Remark 1.4.7.
The above meta theorem now tells us that