Proof of Aut(G): ϕ(Z(G))= Z(G)

[mykayla10]

Homework Statement

For every ϕ in Aut(G), ϕ(Z(G))= Z(G).

Homework Equations

Z(G):={g in G| gh=hg for all h in G}

The Attempt at a Solution

I haven't made too much progress on this one. I know that if I let g be an element of Z(G) that I need to prove that For every ϕ(g) is also and element of Z(G), which means I need to prove that for every h in G ϕ(g)h=hϕ(g). I just do not know where to go from there. I also do not even know where to begin in proving that Z(G) is an element of ϕ(Z(G)) so that I can completely prove the equality.

 

[Dick]

You know that phi is an automorphism. Which means that there is an element j of G such that phi(j)=h. Can you use that to prove phi(g)h=hphi(g)?

 

[Stephen Tashi]

Homework Statement

For every ϕ in Aut(G), ϕ(Z(G))= Z(G).

Homework Equations

Z(G):={g in G| gh=hg for all h in G}

The Attempt at a Solution

I haven't made too much progress on this one. I know that if I let g be an element of Z(G) that I need to prove that For every ϕ(g) is also and element of Z(G), which means I need to prove that for every h in G ϕ(g)h=hϕ(g). I just do not know where to go from there.

Do you mean: I need to prove that $$\phi(Z(G)) \subseteq Z(G)$$?
That involves proving for each $$x \in G$$, $$x \in \phi(Z(G))$$ implies $$x \in Z(G)$$.
Let $$x \in \phi(Z(G))$$ Then there exists a $$g \in Z(G)$$ such that $$x = \phi(g)$$ because $$x$$ is in the image of $$Z(G)$$ under the mapping $$\phi$$. (Now you have your $$\phi(g)$$ to work with.)

Then do the part involving $$h$$.

Since $$\phi$$ s an automorphism, there exists an element $$r$$ such that $$\phi^{-1}(h) = r$$. Show $$gr = rg$$. Then look at $$\phi(rg) = \phi(gr)$$

I also do not even know where to begin in proving that Z(G) is an element of ϕ(Z(G)) so that I can completely prove the equality.

You mean "is a subset".

Let $$g \in Z(G)$$. Look at $$x = \phi^{-1}(g)$$ Show $$x$$ commutes with all elements $$r \in G$$ That proves that $$g$$ is the image of an element in $$Z(G)$$ , so $$g \in \phi(Z(G))$$