Just as we can curry functions, so we can "curry" homomorphisms and actions.
Given an action of group on set , we can consider what happens if we fix the first argument to . Writing for the induced map given by , we can see that is a bijection.
Indeed, we claim that is an inverse map to . Consider . This is precisely , which is precisely . By the definition of an action, this is just , where is the group's identity.
We omit the proof that , because it is nearly identical.
That is, we have proved that is in , where is the Symmetric group; equivalently, we can view as mapping elements of into , as well as our original definition of mapping elements of into .
is a homomorphism in this new sense
It turns out that is a homomorphism. It suffices to show that , where recall that the operation in is composition of permutations.
But this is true: by definition of ; that is because is a group action; that is by definition of ; and that is by definition of as required.