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?

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- Jul 29th 2008, 07:00 AM #1

- Jul 29th 2008, 07:13 AM #2
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