# Proving Find isomorphism: What Else Needed?

• Shackleford
In summary: I'll just do a2 and b2 for now.In summary, the student is trying to find a way to reduce the amount of work needed to show that a function is an isomorphism. He is considering the idea of transforming the matrices, but is unsure if this is the right approach. He is also concerned about whether or not he needs to show the isomorphism for every combination of input matrices.
Shackleford
Is what I did all I need to do? Is there anything else I need to prove?

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110728_175901.jpg?t=1311905852

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110728_175917.jpg?t=1311905865

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Hi Shackleford!

You're probably not going to like this, but you're not done yet. You have yet to show that this is an isomorphism.

Your function f is an isomorphism if it is bijective (this is trivial) and if

$$f(xy)=f(x)f(y)$$

for all x and y. You have to show that this holds for your groups.

For example, you'll need to show that

$$f(a^2)=f(a)^2,~f(ba)=f(b)f(a)$$

and so on for every element. This is a lot of work, but maybe you can find some shortcuts that could reduce the calculations a bit? Or maybe you won't bother with checking all of these things

micromass said:
Hi Shackleford!

You're probably not going to like this, but you're not done yet. You have yet to show that this is an isomorphism.

Your function f is an isomorphism if it is bijective (this is trivial) and if

$$f(xy)=f(x)f(y)$$

for all x and y. You have to show that this holds for your groups.

For example, you'll need to show that

$$f(a^2)=f(a)^2,~f(ba)=f(b)f(a)$$

and so on for every element. This is a lot of work, but maybe you can find some shortcuts that could reduce the calculations a bit? Or maybe you won't bother with checking all of these things

Am I correct in making the transformation by assigning the matrices e, a, b, and ab respectively? If so, then I already showed ab = [ ] [ ] = ba. I need to do a2, too?

Shackleford said:
Am I correct in making the transformation by assigning the matrices e, a, b, and ab respectively? If so, then I already showed ab = [ ] [ ] = ba. I need to do a2, too?

Yes, you're matrices are assigned perfectly! The assignment you wrote down is an homomorphism, but you just need to prove it.

One thing to consider is that both groups are abelian. This means that you don't need to check ab and ba. You just need to check one. So I think there's really only 6 mappings you need to check and it is better to do it in a certain order, I think.

Robert1986 said:
One thing to consider is that both groups are abelian. This means that you don't need to check ab and ba. You just need to check one. So I think there's really only 6 mappings you need to check and it is better to do it in a certain order, I think.

I need to check six mappings?

The properties for isomorphism are one-to-one correspondence and, generally, if φ(a*b) = φ(a) x φ(b).

Clearly it has the one-to-one correspondence, and I've already shown φ(ab) = φ(a)φ(b). As mentioned previously, I suppose I need to do a2 and b2.

Don't you need to show it for a*ab b*ab and ab*ab. These will be trivial, but I don't see how you can get around it.

Robert1986 said:
Don't you need to show it for a*ab b*ab and ab*ab. These will be trivial, but I don't see how you can get around it.

I suppose you're right, since the set is so small.

## 1. What is an isomorphism?

An isomorphism is a mathematical concept that describes a relationship between two structures that preserves their underlying properties. In graph theory, an isomorphism between two graphs means that the two graphs have the same structure, even if their vertices and edges are labeled differently.

## 2. How do you prove that two graphs are isomorphic?

To prove that two graphs are isomorphic, you must find a mapping between their vertices that preserves the structure of the graphs. This means that vertices in one graph are mapped to vertices in the other graph in such a way that the number of edges and their connections remain the same.

## 3. What techniques are used to prove isomorphism?

There are several techniques that can be used to prove isomorphism, including the adjacency matrix method, the degree sequence method, and the path and cycle method. These techniques involve examining the structural properties of the graphs and finding similarities between them.

## 4. What additional information is needed to prove isomorphism?

In addition to the two graphs being compared, you will also need to know the number of vertices and edges in each graph. This information is necessary to determine if there is a one-to-one mapping between the graphs that preserves their structure.

## 5. How is proving isomorphism useful in science?

Proving isomorphism is useful in various scientific fields, including chemistry, biology, and computer science. In chemistry, it can be used to identify molecules with the same structure but different arrangements of atoms. In biology, it can help identify similar genetic structures in different organisms. In computer science, it is used for data compression and encryption.

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