Answer all questions

- 1.
- (a)
- Let
*G*be a finite group of order*m*, and let*p*be the smallest prime which divides*m*. Prove that if*H*is a subgroup of index*p*, then*H*is a normal subgroup of*G*. - (b)
- Prove that any group of order
*p*^{ 2}*q*is solvable, where*p**q*are primes. (Hint: consider separately the cases*p*>*q*and*p*<*q*).

- 2.
- List all groups of order 6 (up to isomorphism), and prove that they are the only ones.
- 3.
- If
*A*and*B*are finitely generated abelian groups with*A**A*isomorphic to*B**B*, prove that*A*and*B*are isomorphic. - 4.
- Suppose
0
*A**B**C*0 is a short exact sequence of modules over a ring*R*. Prove that if the sequence splits, i.e. there is an*R*-module homomorphism*h*:*C**B*such that*gh*= 1_{C}, then*B**A**C*. - 5.
- Let
*R*be a commutative ring with a 1. If*S*is a multiplicative set (i.e.*x*,*y**S**xy**S*) containing 1, but not 0, prove there exists a prime ideal of*P*of*R*with*P**S*= . - 6.
- Let
*A*be an abelian group and let*m*> 1 be an integer. Prove that*A*/*m**A*/*mA*. - 7.
- Let
*K*/*k*be a Galois extension of fields and let*f*(*x*)*k*[*x*] be an irreducible polynomial which has a root in*K*. Prove that*f*(*x*) splits into linear factors in*K*[*x*] . - 8.
- Let
*K*be a finite Galois extension of with Galois group isomorphic to*A*_{4}. For each divisor*d*of 12, how many subfields*L*of*K*have [*K*:*L*] =*d*? In each case give the isomorphism class of (*K*/*L*) , and state whether or not*L*/ is a Galois extension. (Recall*A*_{4}is a counter-example to the converse of Lagrange's theorem.)