Can someone help me with proving this proposition:
Let a group. For we define:
Proof is an automorphism of (We call it an inside automorphism of determined by ).
If I wan't to proof is an automorphism I've to demonstrate that is an isomorphism or an bijective homomorphism.
1) is an homorphism
But to prove if is an homorphism in my inion I need a composite law for the group .
I guess I've to proof for two arbitrary chosen numbers in that:
But how do I now: ? Or doesn't that matter? ...
2) is bijective:
I've to prove that:
In this case:
This is true because in a group a is the symmetric element of and so:
So the only thing I haven't proved yet is the homorphism. Does someone have a hint?
Thanks in advance! .
(Sorry If my English wouldn't be correct, but English mathematical terminology is not easy for me as a Belgian).