# Mapping an isomorphism b/w 2 grps

by Bachelier
Tags: b or w, grps, isomorphism, mapping
 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?
 P: 376 also does an isomorphism maps connected (separated) sets to connected (separated) sets?
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PF Gold
P: 3,288
 Quote by Bachelier 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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PF Gold
P: 3,288
Mapping an isomorphism b/w 2 grps

 Quote by Bachelier 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?
P: 376
 Quote by jbunniii 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?
Mentor
P: 18,334
 Quote by Bachelier 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?
 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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