Attempting to better understand the group isomorphism theorm

In summary, the homomorphism p:G-->H induces an isomorphism between G/Ker(p) and H if p is onto. This is due to the fact that if p(a) = p(b), then aK = bK in the factor group. This can be understood as the inverse image of p(a) being equal to aK, and the inverse image of p(b) being equal to bK. This concept is more fundamental than group theory and is related to basic sets and functions.
  • #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
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
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
how does aK = p(-1)^(p(a))? wouldn't p^(-1)(p(a)) = a? applying the mapping and then the inverse mapping?
 
  • #5
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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1. What is the group isomorphism theorem?

The group isomorphism theorem is a fundamental concept in abstract algebra that states that two groups are isomorphic if and only if they have the same structure, meaning that they can be mapped onto each other in a way that preserves the group operation and structure.

2. Why is understanding the group isomorphism theorem important?

Understanding the group isomorphism theorem is important because it allows us to identify and classify groups that have the same structure, making it easier to study and analyze them. It also helps us to better understand the relationships between different groups and how they are related to each other.

3. How does the group isomorphism theorem relate to other mathematical concepts?

The group isomorphism theorem is closely related to other mathematical concepts such as homomorphisms, automorphisms, and subgroups. It also has applications in other areas of mathematics such as number theory, geometry, and topology.

4. What are some examples of groups that are isomorphic?

Some examples of groups that are isomorphic include the cyclic group of order 4 and the Klein four-group, the symmetric group of order 3 and the dihedral group of order 6, and the additive group of integers and the multiplicative group of nonzero rational numbers.

5. How can the group isomorphism theorem be used in practical applications?

The group isomorphism theorem has many practical applications, particularly in computer science and cryptography. It is used in the design of error-correcting codes, data compression algorithms, and encryption techniques. It is also used in physics and chemistry to understand the symmetries of molecules and crystals.

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