Do all problems

- Let
*G*be a simple group of order 240 = 2^{4}·3·5.- (a)
- Prove that
*G*has a subgroup of order 15, and that all groups of order 15 are cyclic. - (b)
- Prove that
*G*has exactly 32 elements of order 3. (Hint: show 15 divides | N_{G}(*P*)| where*P*is a Sylow 3-subgroup.)

- Let
*R*be a UFD with the property that any ideal that can be generated by two elements is principal.- (a)
- If
*I*_{1}⊆*I*_{2}⊆... is an ascending chain of principal ideals in*R*, prove that there exists*N*∈ℕ such that*I*_{n}=*I*_{N}for all*n*>*N*. - (b)
- Prove that
*R*is a PID.

- Let
*R*be a ring with a 1 and let*S*be a subring of*R*with the same 1. Prove or give a counterexample to the following statements.- (a)
- If
*P*is a projective left*S*-module, then*R*⊗_{S}*P*is a projective left*R*-module. - (b)
- If
*P*is an injective left*S*-module, then*R*⊗_{S}*P*is an injective left*R*-module.

- Let
*R*be a PID and let*M*and*N*be finitely generated*R*-modules. Suppose that*M*^{3}≌*N*^{2}. Prove that there exists an*R*-module*P*such that*P*^{2}≌*M*. - Let
*k*be an algebraically closed field of characteristic 2 and let*A*be a square matrix over*k*such that*A*is similar to*A*^{2}.- (a)
- Prove that
*A*is similar to*A*^{2n}for all positive integers*n*. - (b)
- Prove that
*A*is similar to a diagonal matrix over*k*.

- Explicitly construct a subfield
*K*of ℂ such that [*K*: ℚ] = 3 and*K*is Galois over ℚ. For such a field*K*, prove that*K*(√2) is a Galois extension of ℚ, and determine the Galois group Gal(*K*(√2)/ℚ). - Let
*H*be a central subgroup of the finite group*G*and let χ be a character of*H*. Assume that*H*≠*G*. Prove that Ind_{H}^{G}(χ) (the induced character) is not an irreducible character of*G*(all characters are assumed to be over ℂ).

Peter Linnell 2016-08-25