Apologies, the LaTeX thing doesn't seem to be working, so not very clear!(adsbygoogle = window.adsbygoogle || []).push({});

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

1. The problem statement, all variables and given/known data

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)

2. Relevant equations

Unknown

3. 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.

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# Homework Help: Group/Representation Theory Help! (Summations and elements of a finite group)

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