Prove Beta is an isomorphism of groups

In summary, the conversation is about the concepts of 1-to-1 and onto and how they can be applied in proving the isomorphism of groups. The participants also discuss an example problem to help understand the concept better.
  • #1
SqrachMasda
42
0
i can't grasp these concepts, 1-to-1 and onto have always annoyed me.

here's 1 question, (i don't know how to post symbols so Beta ..)
(C is Complex numbers)
Let Beta:<C,+> -> <C,+> by Beta(a+bi)=a-bi (that is, the image is a +(-b)i).
Prove Beta is an isomorphism of groups.


i have a lot of these problems but maybe a one may help me understand the rest
 
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  • #2
1-1: f(a+bi)=y
f(c+di)=y ->
f(a+bi)=f(c+di)->a-bi=c-di
->a=c & -bi=-di
->-b=-d->b=d as inverse is unique in integers under addition
so if f(x)=f(n)->x=n

onto: if y is an element of f(x) then y=a-bi for some a,b element of integers. But, c+di is an element of <C,+> for all c, d elements of integers. a, b are integers so a+bi is an element of <C,+> so the function is onto.

f(a+bi + c+di) = a-bi + c-di = f(a+bi) + f(c+di)...
 
  • #3
thanks nnnnnnnn
 
  • #4
don't forget that since f has an obvious inverse map , namely itself, then it must be a bijection, so it suffices to prove it is a homomorphism of additive groups.
 

1. What is an isomorphism of groups?

An isomorphism of groups is a bijective function between two groups that preserves the group structure. This means that the operation of the first group is preserved in the second group, and vice versa.

2. How do you prove that Beta is an isomorphism of groups?

To prove that Beta is an isomorphism of groups, we need to show that it is a bijective function and that it preserves the group structure. This can be done by showing that Beta is one-to-one, onto, and that it preserves the group operation.

3. What does it mean for Beta to be bijective?

Being bijective means that Beta has a one-to-one correspondence between the elements of the first group and the elements of the second group. This ensures that every element in the first group has a unique image in the second group, and vice versa.

4. How does Beta preserve the group structure?

In order for Beta to be an isomorphism of groups, it must preserve the group structure. This means that the operation of the first group must be preserved in the second group. In other words, if we apply the group operation to two elements in the first group, the result should be the same as applying the group operation to the corresponding elements in the second group.

5. Can Beta be an isomorphism of groups if the groups have different structures?

No, Beta cannot be an isomorphism of groups if the groups have different structures. In order for Beta to preserve the group structure, the two groups must have the same operation and elements. If the groups have different structures, Beta may still be a homomorphism, but it cannot be an isomorphism.

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