Im stuck on these problems and could use some help.
1. Show that if p is a representation of a group G, then p* defined as
p*(x) = p(x^-1)^T
is again a representation of G. (Here T means transpose.)
2. Let G be a finite group with normal subgroup N and v be the canonical homomorphism from G to G* := G/N via v : g -> Ng.
Find the kernel of the algebra homomorphism induced by v from F(G) to F(G*) = f(G/N).
Also show that
is an idempotent in F(G).
Show that F(G) - e.F(G) + (1-e)F(G)
with e.F(G) isomorphic to F(G*).