Attempting to better understand the group isomorphism theorm

  • #1
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The homomorphism p:G-->H induces an isomorphism between G/Ker(p) and H (if p is onto). I am trying to understand why this must be true. I understand why these groups have the same magnitude and so a bijection is possible, but there is something that I am not able to understand.

What seems to be true but I don't understand why is that if a and b are elements in G, that if p(a) = p(b) then aK = bK in the factor group. Is this true? It seems like it should be. If so, then can somebody help me understand why it is true on an intuitive level?

Sorry, I realize that this is a broad and poorly phrased question, hoping somebody can see through my lack of competent communication and give me some insight here.
 

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  • #2
lavinia
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Since ##p ## is a homomorphism ##p(ab^{-1}) ## is the identity in H so ##p(ab^{-1}) ## is in the kernel of ## p ##.

So ##ab^{-1} = k## for some ##k ## in K
 
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  • #3
mathwonk
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put another way, aK is exactly equal to p^(-1)(p(a)), since (in combination with lavinia's answer): for x in K, p(ax) = p(a)p(x) = p(a)e = p(a);

so if p(a) = p(b) = y, then aK = p^(-1)(p(a)) = p^(-1)(y) = p^(-1)(p(b)) = bK.
 
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  • #4
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how does aK = p(-1)^(p(a))? wouldn't p^(-1)(p(a)) = a? applying the mapping and then the inverse mapping?
 
  • #5
mathwonk
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there is no inverse mapping. only bijective mappings have inverse mappings, and yours is only surjective. in general, the notation f^(-1), denotes not the inverse mapping from points of the target to points of the source, but rather the inverse image mapping from subsets of the target to subsets of the source.

I.e. in general, as here, if f:S-->T, is a mapping, then f^(-1) of a subset Y in T, means the subset X of S consisting of all points that map into Y.


For a one point subset say y in T, f^(-1)(y) means the subset of all points of S that map to y. For a group homomorphism f:G-->H, the kernel is the subset f^(-1)(e), i.e. the subset of all points of G that map to e, and if y is a point of H, and x is any point of G with f(x) = y, then the coset xK equals the subset f^(-1)(y) of all points of G that map to y.

This material on inverse images of maps that you seem to be missing is more elementary than group theory. Maybe you need to review basic theory of sets and functions.
 
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