is a group. The operation on is defined by for all . Let be the map from G to itself such that where is the inverse of in group . Show that is a group isomorphism from to .

Attempt:

I think I have to show that is one to one, onto and it preserves order:

For "one to one":

For "onto":

I must find some such that .

So

And to show that it's operation preserving I did the following:

I appreciate it if anyone could correct any of my mistakes.