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 gP and xX, 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 HG 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 HG.)

3. Let R be a noetherian UFD with the property whenever x1,..., xnR such that no prime divides all xi, then x1R + ... + xnR = 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 Mℤ/qℤ = 0.

5. Let M be a finitely generated ℂ[x]-module. Suppose there exists a submodule N of M such that NM and NM. Prove that there exists c∈ℂ such that (x -c)MM and (x -c)MM.

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.

Peter Linnell 2014-08-06