In this topic we’ll solve following problem:
Let be an irreducible representation of of degree and character ; let be the center of and let be its order.
(a)Show that is a homothety for each . Deduce from this that for all .
(b)Prove the inequality .
(c)Show that, if is faithful, the group is cyclic.
The problem is posted in the website mathscope.org by PnAT and solved by 2M and Vodka, they are members of the site too. This is it in detail:
(a)For each and we have . Therefore by Schur’s lemma we have is a homothety. Now for all put , since we have is a root of unit, particularly . Final, .
(b)This part is easy! In fact, we have and we’re done.
(c)If is faithful then by all what above we can see as a subgroup of the group th roots of unit, and therefore is cyclic.