Remember Isomorphism Theorems: Intuition Guide

[lion8172]

Does anybody know of a nice, intuitive way to remember the second and third isomorphism theorems?

 

[quasar987]

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.

 

[morphism]

For 2, draw the lattice - for groups it looks something like this:

     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$\cap$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.