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Group homomorphism

  1. Mar 27, 2007 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant 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...

    3. 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:
     
  2. jcsd
  3. Mar 27, 2007 #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.
     
  4. Mar 28, 2007 #3
    I dont have a homomorphism though... I am not given the fact that Ф is a homomorphism????
     
  5. Mar 28, 2007 #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?
     
  6. Mar 28, 2007 #5
    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...
     
  7. Mar 28, 2007 #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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