- 1.
- Let
*p*be a prime, let*G*be a finite*p*-group, let*Z*be the center of*G*, and let 1*H**G*. (i) Let*x**H**Z*, and let (*x*) denote the conjugacy class containing*x*. Prove that (*x*)*H*and*p*divides |(*x*)|.(ii) Prove that

*Z**H*1.(iii) Let

*A*be a maximal normal abelian subgroup of*G*. Prove that*A*is also a maximal abelian subgroup of*G*. (Apply (ii) with*G*=*G*/*A*and*H*the centralizer of*A*in*G*.) **2.**-
Let
*G*be a simple group of order 180.(i) Prove that the number of 5-Sylow subgroups of

*G*is 36.(ii) Prove that the normalizer of a 3-Sylow subgroup of

*G*has order 18.(iii) Prove that the 3-Sylow subgroup of a group of order 18 is normal in that group.

(iv) If

*A*and*B*are distinct 3-Sylow subgroups of*G*, prove that*A**B*= 1 (consider the centralizer in*G*of*A**B*).(v) Prove that there is no simple group of order 180.

**3.**- Let
*R*be a PID (principal ideal domain), and let*M*be a cyclic left*R*-module. Suppose*M*=*A**B*where*A*and*B*are nonzero left*R*-modules. Prove that there exists*r**R*0 such that*rM*= 0. Prove further that for such an*r*, there exist distinct primes*p*,*q**R*such that*pq*divides*r*. **4.**- Let
*R*be the ring [[*X*]]/(*X*- 2), where [[*X*]] denotes the power series ring in*X*over , the localization of at the prime 2.(i) If

*q*is an odd integer, prove that*q*is invertible in [[*X*]]/(*X*- 2).(ii) Define : [[

*X*]][[*X*]] by*a*_{i}*X*^{i}=*a*_{i}*X*^{i}, and let : [[*X*]]*R*be the natural epimorphism. Prove that is surjective and deduce that*R*[[*X*]]/(*X*- 2).(iii) Prove that

*R*[*X*]/(*X*- 2). **5.**- Let
*p*be a prime, let*K**L*be fields of characteristic*p*, let ,*L*, and let*d*be a positive integer. Suppose [*K*() :*K*] =*d*, [*K*() :*K*] =*p*, is separable over*K*, and is not separable over*K*.(i) Prove that

*K*() =*K*() and*K*.(ii) Prove that

*K*()*K*( + ).(iii) Prove that [

*K*( + ) :*K*] =*pd*. **6.**- Let
*p*be a prime, and let*K*= (*t*), the quotient field (field of fractions) of the polynomial ring [*t*].(i) Prove that

*X*^{p}-*t*is irreducible in*K*[*X*].(ii) Let

*L*be the splitting field of*X*^{p}-*t*over*K*. Determine the Galois group of*L*over*K*.