I'm really struggling with this attached homework question. Any help would be greatly appreciated. Thank you.

Let G be a nite group. Dene the group algebra (C[G] to be the vector space over (C with basis {Eg}geg, with the following rule for multiplication: Z \"939 * Z bgeg = Z Z a91b92 '39 (\"9,59 5 C)- gEG 960 96G 91,92EG 9192=9 In particular, eg - eh = egh for all g, h E G. You may use without proof that (C[G] is a ring (with addition being the usual vector space addition, multiplication given as above, and el as the multiplicative identity). (a) Show that 45 : C > C[G] dened by A > A61 is a ring homomorphism. (b) Prove that (C[G] is commutative if and only if G is abelian. (c) Prove that (C[G] is a eld if and only if G is the trivial group