Do all problems

- Let
*p*be a prime, let*H*⊲*G*be finite groups, and let*P*be a subgroup of*G*. Prove that*P*is a Sylow*p*-subgroup of*G*if and only if*P*∩*H*and*PH*/*H*are Sylow*p*-subgroups of*H*and*G*/*H*respectively. - Prove that there is no simple group of order
576 = 9·64.
- Let
*R*be a UFD with exactly two primes,*p*and*q*(i.e.*p*and*q*are nonassociate primes, and any prime is an associate of either*p*or*q*). Given positive integers*m*,*n*, prove that (*p*^{m},*q*^{n}) =*R*(consider*p*^{m}+*q*^{n}). Deduce that*R*is a PID. - Let
*k*be a field, let*M*be a finitely generated*k*[*x*]-module, and let*C*be a cyclic*k*[*x*]-module. Suppose*M*has a proper submodule*N*(so*M*≠*N*) such that*M*≌*N*. Prove that there exists a*k*[*x*]-module epimorphism*M*↠*C*. - Let
*K*and*L*be finite fields, let*K*^{+}indicate the abelian group*K*under addition, and let*L*^{×}indicate the abelian group of nonzero elements of*L*under multiplication. Determine the order of*K*^{+}⊗_{ℤ}*L*^{×}in terms of |*K*| and |*L*|. (You will need to consider two cases, namely whether or not ch*K*divides |*L*^{×}|.) - Let
*K*and*L*be finite Galois extensions of ℚ. Prove that*K*∩*L*is also a finite Galois extension of ℚ. - Let
*G*be a group and let 0→ℤ→*P*→*Q*→ℤ→ 0 be an exact sequence of ℤ*G*-modules (here ℤ is the trivial*G*-module), where*P*and*Q*are projective ℤ*G*-modules. Prove that*H*^{1}(*G*,*X*)≌*H*^{3}(*G*,*X*) for all ℤ*G*-modules*X*.

Peter Linnell 2013-08-23