Group Homomorphism: Prove Existence of Element a

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In summary, the problem asks to prove the existence of an element a in a group G, such that the map @: G⟶G defined by @(x) = a ф(x) for every element x in G, is a homomorphism. The given definitions and properties of homomorphisms and the map ф are used to determine the value of a.
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nadineM
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Homework Statement


Let G be a group with identity e and product ab for any elements a and b of G.
Let ф: G⟶G be a map such that Ф(a sub1)ф (a sub2)ф(a sub3) = ф(b sub1) ф(b sub2) ф(b sub3) whenever,
(a1) (a2)(a3) = e=(b1) (b2) (b3) for any(not necessarily distinct) elements a1 ,a2 ,a3, b1, b2, b3 of G.
Prove: There exists an element a in G such that the map @: G⟶G defined by @(x) = a ф(x) for every element x in G, is a homomorphism.

Homework Equations


Def: A homommorphism Ф from a group G to a group G is a mapping from G to G that preserves the group opperation. That is, Ф(ab) = Ф(a)Ф(b) for all a, b in G
Properties of elements under a homomorphism:
Ф carries the identity in G to the identity in G
Ф preserves inverses
*note that the Ф in this section is not the same as in the question...

The Attempt at a Solution



I really haven't got a clue on where to even begin to define "a" I am thinking that it needs to be triplet for example aea^(-1)...but I have no idea what to do or where to start...I am completely lost, can anyone give me a push in the right direction?:confused:
 
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  • #2
Combine the facts "homomorphisms carry the identity in G to the identity in G" with "@(x) = aф(x)" to determine a.
 
  • #3
I don't have a homomorphism though... I am not given the fact that Ф is a homomorphism?
 
  • #4
AKG's point is that '@' is supposed to be a homomorphism, not the original function. So @(e) had better be e. Can we change symbols here?
 
  • #5
Dick said:
AKG's point is that '@' is supposed to be a homomorphism, not the original function. So @(e) had better be e. Can we change symbols here?

OK thanks...I will work with that for a bit and see where I get...you can use what ever symbol you like i tried using gamma but couldn't get it to come up right...
 
  • #6
It might help to experiment with ф to try and understand it better.

You have an identity it satisfies -- try plugging special values into those identities to see if you can deduce other facts about ф.

Try choosing a group and constructing an actual function ф that satisfies the listed properties.
 

1. What is a group homomorphism?

A group homomorphism is a function that preserves the structure of a group. In other words, it maps elements from one group to another in a way that respects the group operation. This means that if you apply the function to two elements in the first group and then combine the results using the group operation, it will be the same as combining the two elements first and then applying the function.

2. How do you prove the existence of an element a in a group homomorphism?

To prove the existence of an element a in a group homomorphism, you must show that the function maps the identity element of the first group to the identity element of the second group. This ensures that the function is well-defined and there is an element a that is mapped to the identity element.

3. Why is it important to prove the existence of an element a in a group homomorphism?

Proving the existence of an element a in a group homomorphism is important because it ensures that the function is valid and can be used to map elements between groups. It also guarantees that the group operation is preserved, which is crucial in many mathematical and scientific applications.

4. Can you give an example of a group homomorphism?

One example of a group homomorphism is the function f(x) = 2x, which maps elements from the group of integers (under addition) to the group of even integers (under addition). This function preserves the group operation, as adding two even integers will always result in an even integer, and it maps the identity element (0) to the identity element (0).

5. How does the existence of an element a in a group homomorphism relate to the concept of isomorphism?

The existence of an element a in a group homomorphism is necessary for the function to be an isomorphism. An isomorphism is a bijective homomorphism, meaning it is a one-to-one and onto function that preserves the group operation. To prove that a group homomorphism is an isomorphism, you must show that there exists an element a that is mapped to the identity element in both groups.

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