Algebra Prelim, August 2016

Do all problems

1. Let G be a simple group of order 240 = 2⁴·3·5.

   (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.)

2. Let R be a UFD with the property that any ideal that can be generated by two elements is principal.

   (a)

   If I₁⊆I₂⊆... 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.

3. 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⊗_SP is a projective left R-module.

   (b)

   If P is an injective left S-module, then R⊗_SP is an injective left R-module.

4. Let R be a PID and let M and N be finitely generated R-modules. Suppose that M³≌N². Prove that there exists an R-module P such that P²≌M.

5. 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 A².

   (a)

   Prove that A is similar to A²ⁿ for all positive integers n.

   (b)

   Prove that A is similar to a diagonal matrix over k.

6. 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).

7. 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).