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OMM!

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I am working through a book on representation theory, but am stuck on these exercises.

## Homework Statement

Let G be a finite group and let E = Sum g (running over all g in G).

(i) Prove: Ex = E (forall x in G)

(ii) Prove: E^{2} = nE (|G| = n)

(iii) Find 3 e_{i} in G s.t. (e_{i})^{2} = e_{i} (forall i = 1,2,3)

## Homework Equations

Unknown

## The Attempt at a Solution

(i) In order to show that Ex = E, we must show that there is a bijection that maps from G to G which sends g to gx for all g in G. Must show that this map is surjective and injective to show it is bijective. Little bit stuck on showing these.

(ii) I am told that part (ii) requires part (i). Am I correct in thinking that E^{2} = (Sum {running over elements of G} (Sum {running over elements of G} g))? Basically the sum of the sum! (Sorry LaTeX working would make this far easier to explain!)

(iii) Again, part (iii) requires part (ii), but I'm not even quite sure what I'm being asked to show. Is it simply that there are 3 elements in the finite group G that when squared are equal to themselves?

Thanks for any help.