Let G be a finite commutative group, Let n in Z with (|G|,n)=1. Show that the function f:G to G defined by f(g)=g^n, all g in G. Prove that it is isomorphism?
We'll prove that is bijective and that
Since and then (1) where is the identity element. Because the order of divides
Now suppose we have but by definition of Group and by (1) it follows that thus this means it's injective
Now let where iff and
Then the set has different elements in G, because f is injective, thus it follows that and so it's surjective. Thus f is bijective
Now obviously thus it is an Isomorphism