Exercises #1 (corrected)

I'm putting these here so you can check that you're keeping up. If you find something weird, or want a (nother) hint, post a comment. (If you've got a hint, you can post that too.) Or you can just come and ask me about them after class.

1. Let V be a finite-dimensional irrep of an abelian group A (possibly infinite, even infinitely generated!). Show that V is 1-dimensional.
Hint: either every group element acts a multiple of the identity, or some element doesn't.

2. Let G be a finite abelian group. Show that the number of distinct irreps of G is |G|. (Hint: start with the cyclic case.)

3. Let G be a group, and Rep(G) resp. Irr(G) the set of {reps resp. irreps of G up to isomorphism}. Let Pic(G) be the set of {1-d reps of G up to isomorphism}. Show that Pic(G) is a group under tensor product (which includes checking some well-definedness), and that Pic(G) acts on Rep(G) by tensor product, preserving the subset Irr(G).

4. Calculate Pic(S₃), and find an element of Rep(S₃) that is fixed under the action from #3.

5. Generalizing #2 & #3, show that Irr(G)Pic(G) is isomorphic to G/[G,G]. Warning: there is no natural isomorphism!