Algebra Prelim, August 2020

Do all problems

1. Classify groups of order 63 up to isomorphism.

2. Prove that any invertible matrix A with complex entries has a square root (i.e., there exists a matrix B such that A = B²).

   (Hint: Use the Jordan canonical form.)

3. Let R be a PID. Determine, up to isomorphism, all finitely generated R-modules A and B whose tensor product A⊗_RB is a free R-module (possibly 0).

4. Suppose A is a finite abelian group and p is a prime dividing its order. Let pⁿ be the largest power of p dividing | A|. Prove that Hom(ℤ/pⁿℤ, A) is isomorphic to the (unique) Sylow p-subgroup of A.

   (Note: Hom(ℤ/pⁿℤ, A) is the group of all homomorphisms f : ℤ/pⁿℤ→A.)

5. Let E be the splitting field of x³ - 4 over ℚ. Find, with proof:

   (a)

   the Galois group and all intermediate fields of E/ℚ;

   (b)

   θ∈E such that E = ℚ(θ).

6. Let R be a commutative ring with 1≠ 0. Suppose a∈R is an element such that aⁿ≠ 0 for all n > 0.

   (a)

   Let 𝓢 be the set of all ideals I of R such that aⁿ∉I for all n≥ 0. Show that 𝓢 satisfies the conditions of Zorn's lemma with respect to inclusion.

   (b)

   Use (a) to prove that there exists a prime ideal P of R such that a∉P.

7. Let G be a finite group with subgroups H, K < G. Consider the left action of the direct product H×K on G given by (h, k)·x = hxk^-1.

   (a)

   Show that the (H×K)-orbit of x∈G has | H×K|/| H∩xKx^-1| elements.

   (b)

   Consider the special case of the above where G = GL₂(𝔽_q), the group of invertible 2×2 matrices over a finite field 𝔽_q (with |𝔽_q| = q), and

   H = K = {

   (

   	a	c
   	0	b

   )

   : a, b, c∈𝔽_q, a, b≠0} < G.

   Show that the (H×H)-action on G has two orbits, and compute the size of each orbit. (You may use without proof that | G| = (q² -1)(q² -q).)