Abstract Algebra: need a review of 1-1 and onto proof

[Halaaku]

Homework Statement

define a function f--> gHg^{-1}

Homework Equations

prove if f is 1-1 and onto.

The Attempt at a Solution

1-1:
f(h1)=f(h2)
gh1g^{-1}=gh2g^{-1}
h1=h2 (left and right cancellations)

onto:
f(g^{-1}hg)=gg^{-1}hgg^{-1}=h
so every h belonging to H has an image of g^{-1}hg.
However,I do not really understand the last line. I followed the example that we did in class but now I am not sure.

 

[pasmith]

Homework Statement

define a function f--> gHg^{-1}

If you tell me what H and g are (I'm assuming there's a group G, that H is a subgroup of G, and that g \in G) I will know the domain and codomain of f, but you haven't said anything about what f(h) is for h \in H, so I can't say whether f is injective or surjective.

Homework Equations

prove if f is 1-1 and onto.

The Attempt at a Solution

1-1:
f(h1)=f(h2)
gh1g^{-1}=gh2g^{-1}
h1=h2 (left and right cancellations)

That's one mystery solved: f: h \mapsto ghg^{-1} and it is indeed 1-1.

onto:
f(g^{-1}hg)=gg^{-1}hgg^{-1}=h
so every h belonging to H has an image of g^{-1}hg.

No. What you've written here is that if g^{-1}hg \in H then f(g^{-1}hg) = h \in gHg^{-1}. But unless H is normal, and you haven't told me that it is, it may not be the case that if h \in H then g^{-1}hg \in H.

To prove that f is onto, you need to start with a k \in gHg^{-1} and show that there exists h \in H such that f(h) = k.