Group homomorphism

  • Thread starter 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 havent 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:
 

Answers and Replies

  • #2
AKG
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Combine the facts "homomorphisms carry the identity in G to the identity in G" with "@(x) = aф(x)" to determine a.
 
  • #3
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I dont have a homomorphism though... I am not given the fact that Ф is a homomorphism????
 
  • #4
Dick
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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
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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 couldnt get it to come up right...
 
  • #6
Hurkyl
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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.
 

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