I'm going to have to chew on this some more before I try the exercises. I'm not clear on a couple of things here:
What is the difference between φ[A] and φ(A)? It seems as if they would give the same result.
Proving the first item of the first theorem, I'm not sure I understand the algebra. Multiplying both sides of φ(e) = φ(e)φ(e) by φ⁻¹(e), I get φ(e)/φ(e) = φ(e)φ(e)/φ(e), and I don't follow why the left side cancels to e' while the φ(e)/φ(e) on the right just goes away. I get that e^2=e, and I can see e/e=e, but I'm not clear on exactly where the e' came from.
I think the reason this notation is used for φ[A] and φ(A) is to differentiate when we're thinking about an individual object and a set.
The fractions both cancel to e', on the left and on the right. So we get φ(e)/φ(e) = φ(e)φ(e)/φ(e), but φ(e)/φ(e) = e' (by the definition of the inverse - remember that φ(e) is an object in G'). So the LHS is e', and the RHS is φ(e) • e' = φ(e). Does that clear things up?
Thanks, yes. I had a feeling φ[A] versus φ(A) might have to do with the set versus its members. I did end up with e' = φ(e) • e', but it somehow ended up feeling more like a tautology than a proof. Maybe because φ(e) = e' is given by the homomorphism?
I'm going to have to chew on this some more before I try the exercises. I'm not clear on a couple of things here:
What is the difference between φ[A] and φ(A)? It seems as if they would give the same result.
Proving the first item of the first theorem, I'm not sure I understand the algebra. Multiplying both sides of φ(e) = φ(e)φ(e) by φ⁻¹(e), I get φ(e)/φ(e) = φ(e)φ(e)/φ(e), and I don't follow why the left side cancels to e' while the φ(e)/φ(e) on the right just goes away. I get that e^2=e, and I can see e/e=e, but I'm not clear on exactly where the e' came from.
I think the reason this notation is used for φ[A] and φ(A) is to differentiate when we're thinking about an individual object and a set.
The fractions both cancel to e', on the left and on the right. So we get φ(e)/φ(e) = φ(e)φ(e)/φ(e), but φ(e)/φ(e) = e' (by the definition of the inverse - remember that φ(e) is an object in G'). So the LHS is e', and the RHS is φ(e) • e' = φ(e). Does that clear things up?
Thanks, yes. I had a feeling φ[A] versus φ(A) might have to do with the set versus its members. I did end up with e' = φ(e) • e', but it somehow ended up feeling more like a tautology than a proof. Maybe because φ(e) = e' is given by the homomorphism?