Originally Posted by

**Deveno** 1. we need to show that p* is a homomorphism (it should be clear that the matrix p(x^-1)ᵀ is in GLn(F) if p(x) is).

so: p*(xy) = [p((xy)^-1)]ᵀ = [p(y^-1x^-1)]ᵀ = [p(y^-1)p(x^-1)]ᵀ (since p is a homomorphism)

= [p(y)^-1p(x)^-1]ᵀ = (p(x^-1))ᵀ(p(y^-1))ᵀ = p*(x)p*(y)

2. is F(G) the group algebra of G over F? if so then the induced homomorphism (which i shall call v') takes Σajgj to Σajv(gj).

so ker(v') = {Σajgj in F(G) : Σajv(gj) = v(e) = N}. this implies that v(gj) = N for all j (so gj is in N), and that Σaj = 1.