Group action induces homomorphism to the symmetric group

https://arbital.com/p/group_action_induces_homomorphism

by Patrick Stevens Jun 14 2016 updated Jun 14 2016

We can view group actions as "bundles of homomorphisms" which behave in a certain way.


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.