Group Homomorphism: Prove Existence of Element a

[nadineM]

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?

 

[AKG]

Combine the facts "homomorphisms carry the identity in G to the identity in G" with "@(x) = aф(x)" to determine a.

 

[nadineM]

I don't have a homomorphism though... I am not given the fact that Ф is a homomorphism?

 

[Dick]

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?

 

[nadineM]

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...

 

[Hurkyl]

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.