Showing Two Groups are Isomorphic

[Oxymoron]

How would I show that two groups are isomorphic?

FOR EXAMPLE:

Take the group homomorphism φ : ((0, oo), x) → ((0, oo), x) defined by φ (x) = x²

Since φ is taking any element in (0, oo) and operating on it by x, does it map one-to-one and onto to (0, oo)?

I assume by showing that two groups are isomorphic you have to show that there is a one-to-one correspondence and that they are onto (ie. the two groups are a bijection).

Would I start by taking some element a of ((0, oo), x) and then say that under x, a is mapped to a². Then for all a, a² is in (0, oo) hence it is one-to-one. Then show that there is only one a that maps to a² in (0, oo) hence it is onto. Since it is both then it is isomorphic.

Any help would be appreciated.

 

[gravenewworld]

To show a mapping is isomorphic you must show 2 things (ok really 3 things), that the mapping is is a bijection and that is preserves operation.


To show that it is 1-1 assume that P(a)=P(b) (P is phi). Prove that a=b.

To show that it is onto you must show that for any element g* in G* (G* is the group that G maps to) there exists a g in G that gets mapped to it, i.e. P(g)=g* .

Finally show that the mapping preserves operation, that is P(ab)=P(a)P(b) for all a,b in G.

 

[matt grime]

well, you've two xs in that post that mean different things.

firstly, the map must be a homomorphism of groups, you've not shown that squaring is a homomorphism from the positive numbers under * to itself. then it suffices to show this map is a bijection which is trivial (since it has an inverse).

 

[Oxymoron]

Matt, are you saying that if the map has an inverse then it is automatically a bijection. In other words, you can either show 1-1 and onto OR show that it has an inverse?



Here is my proof that the two groups are isomorphic...

To show that φ is 1-1 assume φ(a) = φ(b) for any a,b in (0,oo). Prove a = b.

Take a,b in (0, oo) then φ(a) = a², φ(b) = b²

If φ(a) = φ(b) then a² = b² which implies that a = b.
Hence φ is 1-1

To show that φ is onto show that for any g* in (0, oo) there exists a g in (0, oo) such that φ(g) = g*

Take g in (0,oo) then φ(g) = g² = g* since then g* is going to be in (0, oo) because it is a homomorphism under multiplication, then φ is onto.

Finally, it preserves operation since taking any a in (0, oo)
φ (ab) = (ab)² = a²b² = φ(a)φ(b)

Since φ is 1-1 and onto then the two groups (0, oo) and (0, oo) are isomorphic.

 

[matt grime]

a map is a bijection iff it has an inverse, that is a trivial fact, one I'm surprised you've not seen if you're doing group theory.

 

[Oxymoron]

However, if I had a function

φ : (Z, +) → (Z, +) defined by φ (n) = n²

This is NOT homomorphic is it? Since for any a,b in Z

φ(a+b) = (a+b)² = a² + b² + 2ab

which does not equal φ(a)+φ(b)

Does this mean automatically that the two groups are not isomorphic?

 

[Muzza]

To show that φ is onto show that for any g* in (0, oo) there exists a g in (0, oo) such that φ(g) = g*

Take g in (0,oo) then φ(g) = g² = g* since then g* is going to be in (0, oo) because it is a homomorphism under multiplication, then φ is onto.

Surely you mean "Take g in (0, oo) such that g = sqrt(g*)".

Does this mean automatically that the two groups are not isomorphic?

No, it just means that the function named phi you described is not an isomorphism. There are more functions from (Z, +) to (Z, +) than phi(n) = n^2, the proof that phi(n) = n^2 is not an isomorphism obviously says nothing about the (possible) non-existance of other functions which /are/ bijective homomorphisms (they exist though, f(n) = n will work).

 

[Oxymoron]

Matt, I am not doing group theory. I am doing linear algebra but for some reason the faculty decided to throw in a two-week crash course in elementary abstract algebra.

 

[Oxymoron]

Muzza, correct. I omitted the next step.

 

[Muzza]

Muzza, correct. I omitted the next step.

Just FYI, the omitted step is crucial and your argument is not valid without it ;)

 

[Oxymoron]

Muzza, if g = √(g*) then g is not in Z since Z is the set of integers. √(g*) is not always an integer. Could you explain? I thought it was isomorphic on the condition that g be back in Z?

 

[Muzza]

But we aren't in Z anyway, we're in (0, oo) which I figured was the positive reals (and all positive reals have a real positive square root). If (0, oo) denotes the positive integers (which would be very strange), then ((0, oo), *) isn't a group anyway (what would the inverse of 2 be?)...

*edit* Replaced "rationals" with "integers".

 

[Oxymoron]

R OMG! I am so stupid. I got mixed up between my two questions. Sorry about that.

Well, then that makes perfect sense then!

If we changed the set we where working on to the reals...

φ : (R, +) → (R, +) defined by φ(x) = 2x

This would be a group homomorphism because it preserves addition
For any a,b in R we have

φ (a+b) = 2(a+b) = 2a + 2b = φ (a) + φ (b)

And the identity is mapped back into R...

φ (0) = 2(0) = 0

This is also an isomorphism by a similar argument.

I think I am getting the hang of this...

 

[gravenewworld]

Be extremely careful when working with groups. One of the first pitfalls my professor taught me is how to algebraically work with groups. The most common mistake students make when working with a group is that to do (ab)^2=a^2b^2 for a,b elements of some group G. This does not always work, although it might in this case. A better way to show that it preserves operation is by P(ab)=(ab)^2=(ab)(ab)=a(ba)b (by associativity of a group)=a(ab)b (you can permute since the group is abelian)=(aa)bb=a^2b^2 (by associative property again). I know it seems stupid to do it like this, but you have to be picky when working with groups.

 

[matt grime]

However, if I had a function

φ : (Z, +) → (Z, +) defined by φ (n) = n²

This is NOT homomorphic is it? Since for any a,b in Z

φ(a+b) = (a+b)² = a² + b² + 2ab

which does not equal φ(a)+φ(b)

Does this mean automatically that the two groups are not isomorphic?

Just because some map is not an isomorphism does not imply that the two groups are not isomorphic. The identity map between them is obviously an isomorphism in this case.

 

[Oxymoron]

Thanks for the tips Matt and gravenewworld.

This one had me stumped...

det : (GL_2(R), *) → (R \ {0}, X), where * is matrix multiplication. (X is multiplication)

Should I begin by taking two matrices A and B in GL_2(R) and show that det(A*B) = det(A)Xdet(B)?

 

[Muzza]

Should I begin by taking two matrices A and B in GL_2(R) and show that det(A*B) = det(A)Xdet(B)?

I suppose that could be a good start, but you didn't say what you were asked to prove so it's kind of hard to say for sure. ;)

 

[matt grime]

it's certainly a homomorphism presumably you've got to show that the hard way.

clearly it isn't injective, so not an isomorphism, though it is obviously surjective as well (one need only think of diagonal matrices to see why).

 

[Oxymoron]

Sorry about that. You have to show whether or not it is a group homomorphism and decide if it is an isomorphism.

 

[matt grime]

diag{1,x} and diag{x, 1/x}

 

[Oxymoron]

Matt, how did you see that it isn't injective? I am having some difficulty working this out :-C

 

[matt grime]

what's the determinant of a diagonal matrix with entries x and 1/x for any non-zero x?

 

[Oxymoron]

The determinant of a diagonal matrix is just the product of the entries on the main diagonal. So a 2x2 diagonal matrix with diagonal entries x is x². And the determinant of the 2x2 diagonal matrix with entries 1/x is 1/x². Is this what you mean?

 

[Muzza]

The determinant of a 2x2 diagonal matrix with elements x and 1/x (x non-zero) would be 1, not 1/x^2... I.e. there are many matrices which are not equal to one another, but have the same determinant.

Also, you can prove that no isomorphism can exist between those two groups ((R\{0}, x) is abelian but GL_2(R) isn't).

 

[Oxymoron]

I see, I misunderstood. The 2x2 diagonal matrix has entry(11)= x and entry(22) = 1/x with entries(12,21) = 0. Then I agree, the determinant is 1.

But, as you said, many matrices have determinant 1 so the mapping between the two groups is not bijective. Hence it is not isomorphic? You said that because one group is abelian and the other is not indicates that the two groups are not isomorphic, is this another property that I do not know about? Could you explain.

 

[Muzza]

But, as you said, many matrices have determinant 1 so the mapping between the two groups is not bijective. Hence it is not isomorphic?

No, "hence det(x) is not an isomorphism between the two groups" (this might be what you meant though). There is a difference between saying that two groups are isomorphic and saying that a particular function is an isomorphism between two groups.

You said that because one group is abelian and the other is not indicates that the two groups are not isomorphic, is this another property that I do not know about?

Try constructing the proof yourself (proof by contradiction).

 

[Oxymoron]

Why did you choose the diagonal entries to be x and 1/x?

I understand that the identity is mapped to itself
det(I) = 1
And that the group operation is preserved. I can show this by proving
det(A*B) = detA X detB not equal to 0.
Hence the det is a group homomorphism between the two groups.

Thanks for the help guys.

 

[matt grime]

I chose x and 1/x because that was the most obvious way of showing that the map is not injective. Always make your life easier on yourself; diagonal and upper triangular matrices have determinants and traces that are easy to read off.

 

[Oxymoron]

I have another quick question.

If I were asked to prove that a certain operation * was a binary operation on a given set S. Would I simply prove that it obeys closure? That is, take x, y in S and show that x*y is back in S. If so, then it is a binary operator on S.

 

[Muzza]

See the definition of a binary operator (you have to prove that given two elements x and y, x*y is uniquely determined as well).