Is that mapping the homomorphism?

[NasuSama]

Homework Statement

Let G be a group and let Aut(G) be the group of automorphisms of G.

(a) For any g \in G, define \phi_{g}(x) = g^{-1}xg. Check that \phi_{g}(x) is an automorphism.

(b) Consider the map:

\Phi:G \rightarrow Aut(G)
g \mapsto \phi_{g}

Check that \Phi is a homomorphism.

2. The attempt at a solution

Evaluate \Phi at gh with arbitrary x, so:

\Phi(gh) = \phi_{gh}(x) = (gh)^{-1}x(gh) = h^{-1}g^{-1}xgh = \phi_{h}(\phi_{g}(x))
= \phi_{h} \circ \phi_{g} (x) = \Phi(h) \circ \Phi(g)

But the operation is reversed for this situation. So this is considered to be an antihomomorphism.

Any comments?

 

[Fredrik]

In question (a), you probably meant "check that $\phi_g$ is an automorphism". Do you need help with (a)? You have only shown your work for (b).

At the far left, you need to write $\Phi(gh)(x)=\phi_{gh}(x)=$. The x appears out of nowhere in your calculation, and then it magically disappears at the end. So you need to do something similar at the end.

I agree that we end up with $\Phi(gh)=\Phi(h)\circ\Phi(g)$.

 

[Office_Shredder]

The multiplication on the Automorphism group is sometimes written backwards to accommodate this (for example on the wikipedia page about this homomorphism) - where
\left(\phi_{g} \phi_{h}\right)(x) = \phi_h \left( \phi_g(x) \right)
so you should check to make sure that multiplication is composition read from right to left

If this isn't what it's supposed to be then probably there's a type and the homomorphism is supposed to be
g \mapsto \phi_{g^{-1}}

 

[NasuSama]

In question (a), you probably meant "check that $\phi_g$ is an automorphism". Do you need help with (a)? You have only shown your work for (b).

At the far left, you need to write $\Phi(gh)(x)=\phi_{gh}(x)=$. The x appears out of nowhere in your calculation, and then it magically disappears at the end. So you need to do something similar at the end.

I agree that we end up with $\Phi(gh)=\Phi(h)\circ\Phi(g)$.

Nope, don't need help with part (a). Sorry to indicate that beforehand.