Do all problems

- Let
*p*be a prime, let*G*be a finite group, let*P*be a Sylow*p*-subgroup of*G*, and let*X*denote all elements of*G*with order a power of*p*(including 1).- (a)
- Show that
*P*acts by conjugation on*X*(so for*g*∈*P*and*x*∈*X*, we have*g*·*x*=*gxg*^{-1}). - (b)
- Show that {
*z*} is an orbit of size 1 if and only if*z*is in the center of*P*. - (c)
- If
*p*divides |*G*|, prove that*p*divides |*X*|.

- Let
*p*be a prime and let*G*be a finite*p*-group. Prove that if*H*is a maximal subgroup of*G*, then*H*⊲*G*and |*G*/*H*| =*p*. (Hint: use induction on |*G*|, so the result is true for proper quotients of*G*and consider*HZ*. Maximal means*H*has largest possible order with*H*≠*G*.) - Let
*R*be a noetherian UFD with the property whenever*x*_{1},...,*x*_{n}∈*R*such that no prime divides all*x*_{i}, then*x*_{1}*R*+ ... +*x*_{n}*R*=*R*. Prove that*R*is a PID. (Hint: consider gcd.) - Let
*M*be an injective ℤ-module and let*q*be a positive integer. Prove that*M*⊗_{ℤ}ℤ/*q*ℤ = 0. - Let
*M*be a finitely generated ℂ[*x*]-module. Suppose there exists a submodule*N*of*M*such that*N*≌*M*and*N*≠*M*. Prove that there exists*c*∈ℂ such that (*x*-*c*)*M*≠*M*and (*x*-*c*)*M*≌*M*. - Let
*f*(*x*) = (*x*^{5}+*x*^{3}+1)(*x*^{4}+*x*+ 1)∈𝔽_{2}[*x*], and let*K*be a splitting field for*f*over 𝔽_{2}. ( 𝔽_{2}denotes the field with two elements.)- (a)
- Show that
*x*^{2}+*x*+ 1 is the only irreducible polynomial of degree 2 in 𝔽_{2}[*x*]. - (b)
- Let
*K*be a splitting field for*f*over 𝔽_{2}, let α∈*K*be a root of*x*^{5}+*x*^{3}+ 1, and let β∈*K*be a root of*x*^{4}+*x*+ 1. Determine [𝔽_{2}(α,β) : 𝔽_{2}] - (c)
- Determine
Gal(
*K*/𝔽_{2}). (Galois group)

- Compute the character table of
*S*_{3}×ℤ/3ℤ.

Peter Linnell 2014-08-06