Remember Isomorphism Theorems: Intuition Guide

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In summary, the second isomorphism theorem states that if H,K<G and K<N_G(H), then K/(HnK) ~ HK/K, and the third isomorphism theorem states that if N,M are normal in G and N is in M, then (G/N)/(M/N) ~ G/M. To remember this, it is important to remember the formula (G/N)/(M/N) ~ G/M and to understand that N and M must be normal in G and that N is in M for the statement to make sense. For the second theorem, one can draw a lattice to visualize the isomorphism, and for the third theorem, one must remember how fractions work.
  • #1
lion8172
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Does anybody know of a nice, intuitive way to remember the second and third isomorphism theorems?
 
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  • #2
For the second, with hypothesis "If H,K<G, K<N_G(H), then blahblah", draw a picture. Then notice how it "makes sense" that K/(HnK) ~ HK/K, in the sense that HnK is to K what H is to HK.

The third is easy. It says that if N,M and normal in G and N is in M, then the "fraction" (G/N)/(M/N) can be "simplified": (G/N)/(M/N) ~ G/M.

Actally you just have to remember the formula (G/N)/(M/N) ~ G/M because for it to make any sense, we must have that N and M are normal in G and that N in in M, otherwise G/M, G/N and M/n are not defined.
 
  • #3
For 2, draw the lattice - for groups it looks something like this:
Code:
     G
     |
    HK
   /  \\
  H    K
  \\  /
   H[itex]\cap[/itex]K
    |
   {1}
where I'm using \\ to indicate the isomorphism you get when you collapse the line \, namely HK/K =~ H/H[itex]\cap[/itex]K. (Note that when you collapse the other two lines, you get a corresponding statement about indices; what is it?)

For 3, remember how fractions work: (a/b)/(c/b) = a/c.
 

What are isomorphism theorems?

Isomorphism theorems are a set of mathematical theorems that describe the relationship between groups, rings, and modules. They allow us to identify when two mathematical structures are essentially the same, even if they may appear different at first glance.

Why are isomorphism theorems important?

Isomorphism theorems are important because they allow us to simplify complex mathematical structures and make them easier to study. They also help us to understand the underlying structure of a given mathematical object and identify patterns and relationships between different objects.

What is the intuition behind isomorphism theorems?

The intuition behind isomorphism theorems is that they allow us to identify when two objects are structurally equivalent. This means that they have the same underlying structure, even if they may look different on the surface. It is similar to how two puzzles may have different pictures on the front, but the same shape and pieces on the back.

How do I apply isomorphism theorems?

To apply isomorphism theorems, you first need to identify the mathematical structures you are working with, such as groups or rings. Then, you can use the theorems to identify if and how these structures are isomorphic, and thus, simplify your problem. It is important to understand the properties of the structures you are working with in order to correctly apply the theorems.

Are there any limitations to isomorphism theorems?

Yes, there are limitations to isomorphism theorems. They only apply to certain mathematical structures, such as groups, rings, and modules. They also do not provide a complete solution to a problem, but rather a simplified way of looking at the problem. Additionally, isomorphism theorems may not always be easy to apply, as they require a good understanding of the structures involved.

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