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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?
Is every invertible mapping an isomorphism b/w 2 grps or does it have to be linear?
No!Bachelier said:I googled this but couldn't find a clear answer.
Is every invertible mapping an isomorphism b/w 2 grps
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)##.or does it have to be linear?
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 said:also does an isomorphism maps connected (separated) sets to connected (separated) sets?
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?
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?
An isomorphism is a mathematical concept that describes a one-to-one correspondence between two mathematical structures. In the context of groups, an isomorphism is a mapping between two groups that preserves their algebraic structure, meaning it maintains the group operation and identity element.
To determine if two groups are isomorphic, you can look for a bijective mapping between the two groups that preserves the group operation. This means that for every element in one group, there is a corresponding element in the other group and the operation between those elements is the same in both groups.
Mapping an isomorphism between two groups allows us to understand the structure and properties of one group by studying the structure and properties of another group. It also allows us to apply known results and theorems from one group to the other group, making it easier to solve problems.
Yes, it is possible for two groups to have multiple isomorphisms between them. This means that there can be different ways to map one group onto the other while preserving their algebraic structure. However, the number of isomorphisms between two groups is limited by the size and structure of the groups.
Mapping an isomorphism between two groups has many practical applications, especially in the fields of computer science and cryptography. For example, in cryptography, isomorphic groups can be used to create secure encryption and decryption algorithms. In computer science, isomorphic groups can be used to optimize data storage and retrieval algorithms.