Algebra Prelim, August 2014

Do all problems

1. 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|.

2. 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.)

3. Let R be a noetherian UFD with the property whenever x₁,..., x_n∈R such that no prime divides all x_i, then x₁R + ... + x_nR = R. Prove that R is a PID. (Hint: consider gcd.)

4. Let M be an injective ℤ-module and let q be a positive integer. Prove that M⊗_ℤℤ/qℤ = 0.

5. 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.

6. Let f (x) = (x⁵ + x³ +1)(x⁴ + x + 1)∈𝔽₂[x], and let K be a splitting field for f over 𝔽₂. ( 𝔽₂ denotes the field with two elements.)

   (a)

   Show that x² + x + 1 is the only irreducible polynomial of degree 2 in 𝔽₂[x].

   (b)

   Let K be a splitting field for f over 𝔽₂, let α∈K be a root of x⁵ + x³ + 1, and let β∈K be a root of x⁴ + x + 1. Determine [𝔽₂(α,β) : 𝔽₂]

   (c)

   Determine Gal(K/𝔽₂). (Galois group)

7. Compute the character table of S₃×ℤ/3ℤ.