Answer all questions

- Let
*G*be a simple group of order 480 with an abelian Sylow 2-subgroup.- If
*P*and*Q*are distinct Sylow 2-subgroups of*G*, by considering C_{G}(*P**Q*), prove that*P**Q*= 1. - Prove that there is no such group
*G*.

- If
- Let
*R*be a UFD, let*S*be a multiplicatively closed subset of*R*such that 0*S*, and let*p*be a prime in*R*. Prove that*p*/1 is either a prime or a unit in*S*^{-1}*R*. - Let
*k*be the field /2. Classify the finitely generated projective*k*[*X*]/(*X*^{3}+*X*)-modules up to isomorphism. - Let
*R*be a ring, let*M*be a Noetherian*R*-module, and let*J*denote the Jacobson radical of*R*. Prove that either*MJ*^{n}= 0 for some positive integer*n*, or*MJ*^{n + 1}*MJ*^{n}(strict inequality) for all positive integers*n*. - Let
*R*be a nonzero right Artinian ring (with a 1) with no nonzero nilpotent ideals and no nontrivial (0, 1) idempotents. Prove that*R*is a division ring. - Compute the character table of
*S*_{4}. - Let
*K*be a splitting field of the polynomial*X*^{4}- 2 over . Determine the order of Gal(*K*/). Use this to show that*K*contains a subfield*L*such that [*L*: ] = 4 and*L*is normal over .