Prove function is a homomorphism

Jan 2016
4
0
Ohio
Let G be a group. Fix g ∈ G. Define a map φ : G → G by φ(x) = gxg^−1

Prove: φ is an isomorphism

What I Know: I already showed it is bijective. Now, I need help proving the homomorphism part. I know by definition for all a,b in G, f(ab)=f(a)f(b)

Question: How do I show this? For some reason I am getting confused and I don't think it's that difficult but I can't grasp it.

What I Have Done: Let a=gag^-1 and b=gbg^-1. Then f(ab)=f(gag^-1 * gbg^-1) Is this even correct or did I start off totally wrong?

Thanks!!
 
Feb 2012
144
16
What I Have Done: Let a=gag^-1 (...)
you need to prove that f(ab)=f(a)f(b) for all a,b not just for those satisfying the equation a=gag^{-1}

If a,b are any elements of G, f(ab) = g(ab)g^{-1} = ?
 

Country Boy

Math Team
Jan 2015
3,261
899
Alabama
For any a and b, \(\displaystyle f(ab)= gabg^{-1}\), \(\displaystyle f(a)= gag^{-1}\), \(\displaystyle f(b)= gbg^{-1}\). Now, what is f(a)f(b)?
 
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