Mapping an isomorphism b/w 2 grps

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Discussion Overview

The discussion revolves around the nature of isomorphisms between groups, particularly whether every invertible mapping qualifies as an isomorphism and the implications of isomorphisms in the context of connectedness in topological spaces. Participants explore definitions and conditions necessary for a mapping to be considered an isomorphism, as well as the relevance of additional structures like topology.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants assert that an invertible mapping is not necessarily an isomorphism unless it is also a homomorphism, satisfying specific conditions related to group operations.
  • It is noted that a mapping must be both invertible and a homomorphism to qualify as an isomorphism, with references to the necessary properties of the mapping.
  • Questions are raised about the concept of connectedness in relation to isomorphisms, with some participants indicating that connectedness requires additional topological structure beyond group theory.
  • There is confusion expressed regarding the relationship between linear mappings and general invertible mappings, with a participant reflecting on their learning from linear algebra.
  • Participants seek clarification on the types of groups and mappings being discussed, indicating a need for more context to address the questions effectively.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether every invertible mapping is an isomorphism, as there are differing views on the necessary conditions for isomorphisms. Additionally, there is no agreement on the implications of isomorphisms for connectedness without specifying the underlying structures involved.

Contextual Notes

There is a lack of clarity regarding the specific types of groups and mappings being discussed, as well as the topological structures that may be relevant to the questions raised about connectedness.

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


Bachelier said:
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.
 


Bachelier said:
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?
 


jbunniii said:
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?
 


Bachelier said:
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?
 
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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