mapping an isomorphism b/w 2 grps


by Bachelier
Tags: b or w, grps, isomorphism, mapping
Bachelier
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#1
Feb14-13, 07:47 PM
P: 376
I googled this but couldn't find a clear answer.

Is every invertible mapping an isomorphism b/w 2 grps or does it have to be linear?
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Bachelier
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#2
Feb14-13, 08:01 PM
P: 376
also does an isomorphism maps connected (separated) sets to connected (separated) sets?
jbunniii
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Feb14-13, 08:17 PM
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Quote Quote by Bachelier View Post
I googled this but couldn't find a clear answer.

Is every invertible mapping an isomorphism b/w 2 grps
No!
or does it have to be linear?
It has to be invertible AND a homomorphism, meaning it must satisfy ##\phi(ab) = \phi(a)\phi(b)##, where ##\phi## is the mapping and ##a,b## are arbitrary elements of the group. Here, the group operation is written multiplicatively. The additive version is ##\phi(a+b) = \phi(a) + \phi(b)##.

By the way, one might think that it would also be necessary to stipulate that ##\phi^{-1}## is a homomorphism, but that turns out to be automatically true if ##\phi## is a bijection and a homomorphism.

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Feb14-13, 08:20 PM
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mapping an isomorphism b/w 2 grps


Quote Quote by Bachelier View Post
also does an isomorphism maps connected (separated) sets to connected (separated) sets?
Are we still talking about group isomorphisms? There is no notion of "connected" or "separated" for a general group. You need to impose some additional topological structure. So what kind of groups are you working with?
Bachelier
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#5
Feb14-13, 10:48 PM
P: 376
Quote Quote by jbunniii View Post
Are we still talking about group isomorphisms? There is no notion of "connected" or "separated" for a general group. You need to impose some additional topological structure. So what kind of groups are you working with?
The whole question has to deal with analysis..

if A is connected and we have T: A ---> B an isomorphism, can we say T(A) in B is connected?

I guess one still have to show that a mapping is a homomorphism even in analysis. right?
micromass
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#6
Feb14-13, 11:25 PM
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Quote Quote by Bachelier View Post
The whole question has to deal with analysis..

if A is connected and we have T: A ---> B an isomorphism, can we say T(A) in B is connected?

I guess one still have to show that a mapping is a homomorphism even in analysis. right?
OK, but you are clearly not working with just groups. What is the structure you are working with?? What are A and B?? What kind of map is T? It's an isomorphism of what?
Bachelier
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#7
Feb15-13, 08:18 PM
P: 376
I think I was confusing the invertibilty of a Linear Mapping between 2 Vector Spaces with any function that has an inverse.

I remember in my Lin. Alg. course, we learned that if a Linear Transformation T is invertible, then it is an isomorphism between the 2 VS.

Clearly this is not the general case.


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