Let . Then, the question is does hold, for an arbitrary ring?

Clearly, it is enough to show that , that is, .

So, according to my notes, , which I understand perfectly well. However, apparently this is just because . I don't get why that is true!

My thought is perhaps that the action of on elements of a (finite) normal subgroup perhaps induces a unique element of ? That is to say, for , then for ?

Thanks in advance!