The problem asked is in three parts:
For each element of a in the group G, definte Ta: G -> G such that Ta(x) = xa-1 for all x in G
1) Prove Ta is a permutation on G.
(I believe I did this part by showing Ta(x) = Ta(y) implies x = y (1-1). To show whether it is onto, letting y be in G and solve Ta(x) = y for x.
2) Prove that G' = {Ta for all a in G} is a group with respect to mapping composition
I wrote that if Ta is a permutation (I found that it was in the first part), then it is a subset of S, so I need to show it is a subgroup. Is this a fine approach?
3) Define Phi: G ->G' such that Phi(a) = Ta. Determine whether Phi is always an isomorphism.
For this part, I'm trying to show Phi is a homomorphism, 1-1, and onto. I know I could use the kernel to show it is 1-1 as well.
Any help would be appreciated.
For each element of a in the group G, definte Ta: G -> G such that Ta(x) = xa-1 for all x in G
1) Prove Ta is a permutation on G.
(I believe I did this part by showing Ta(x) = Ta(y) implies x = y (1-1). To show whether it is onto, letting y be in G and solve Ta(x) = y for x.
2) Prove that G' = {Ta for all a in G} is a group with respect to mapping composition
I wrote that if Ta is a permutation (I found that it was in the first part), then it is a subset of S, so I need to show it is a subgroup. Is this a fine approach?
3) Define Phi: G ->G' such that Phi(a) = Ta. Determine whether Phi is always an isomorphism.
For this part, I'm trying to show Phi is a homomorphism, 1-1, and onto. I know I could use the kernel to show it is 1-1 as well.
Any help would be appreciated.