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Mapping an isomorphism b/w 2 grps 
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#1
Feb1413, 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? 


#2
Feb1413, 08:01 PM

P: 376

also does an isomorphism maps connected (separated) sets to connected (separated) sets?



#3
Feb1413, 08:17 PM

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PF Gold
P: 3,288

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. 


#4
Feb1413, 08:20 PM

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PF Gold
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Mapping an isomorphism b/w 2 grps



#5
Feb1413, 10:48 PM

P: 376

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? 


#6
Feb1413, 11:25 PM

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#7
Feb1513, 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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