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I am familiar with the first isomorphism theorem, but looking on wikipedia I see this image which I do not understand.

Diagram of First Isomorphism Theorem

A few specific questions about it:

  1. What do the arrows coming from/ leading to a $0$ mean? Are they different from the other arrows? Also, since I clearly don't understand the diagram, I might as well ask what the other arrows mean too.
  2. How am I supposed to read the diagram? Naturally I want to start at the $0$ in the bottom left and go from $\ker f$ to $G/\ker f$.
  3. It seems like the whole statement of the theorem is contained in the one block $G/\ker f \simeq \operatorname{im} f$ so what is the point of the rest of the diagram?
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up vote 6 down vote accepted

You’ll probably find it helpful to read about exact sequences. There are two in this diagram. One is

$$0\longrightarrow\ker f\overset{\kappa}\longrightarrow G\overset{\pi}\longrightarrow G/\ker f\longrightarrow 0\;.$$

Here $0$ is the trivial group, and the exactness of the sequence means that the range of each map is the kernel of the next. Since the range of the first map is just the identity in $\ker f$, $\kappa$ is an injection; it’s actually just the inclusion map from $\ker f$ into the whole group $G$, so its range is $\ker f$ as a subgroup of $G$. This is then the kernel of the quotient map $\pi$. Finally, the range of $\pi$ must be the kernel of the last map, which means that $\pi$ is a surjection (which of course comes as no surprise!).

The other is $0\longrightarrow G/\ker f\overset{\iota}\longrightarrow H$; it says that $\iota$ is an isomorphism, because its kernel is the range of the first map, which is trivial, but it doesn’t say that $\iota$ is a surjective isomorphism.

Then you have the commutative diagram:

$$\begin{array}{c} &&G/\ker f&\\ &\pi\nearrow&&\searrow\iota\\ G&&\overset{f}\longrightarrow&&H \end{array}$$

This is an assertion that $f=\iota\circ\pi$.

As it says in the accompanying discussion, from the existence of the homomorphism $f$ you can infer the existence of everything else in the diagram. For any interpretation beyond that you’re getting into category theory, which I don’t do.

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The arrows from/to $0$ mean a homomorphism, just like the other arrows. Of course, to/from there is only one homomorphsim, so we know which one it is. Moreover, the two diagonal sequences are supposed to be exact, that is the kernel of an imgae equals the image of the previous one. Thus for example $0\to A\to B$ means not more than that $A\to B$ is a monomorphism.

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