Math 171 abstract algebra fall 2014- assignment 7 in this

Question

Math 171: Abstract Algebra, Fall 2014- Assignment 7

1. In this problem we prove the first part of the Sylow Theorem. Suppose G is a group and |G| = p^km with gcd(p, m) = 1. We will prove G has a subgroup of order p^k.

(a) Show that for any integers a ≥ b, mod p, and hence

≡ m mod p.

Hint: Use the Binomial Theorem.

(b) Let Ω = {X ⊆ G | |X| = p^k}.

There are such subsets. Let G act on Ω be left multiplication. That is, for any g ∈ G and X ∈ Ω, g · X = {gx | x ∈ X}. Show that there is some orbit O of this action such that |O| is not divisible by p.

(c) Choose any X ∈ O and let H be the stabilizer of X. Use the Orbit-Stabilizer to prove p^k divides |H|.

(d) Show that |H| ≤ p^k and hence by part (c), |H| = p^k. Conclude the result.

2. (a) Let ? be any set and suppose G acts on ?. Further suppose |G| = p^k for some prime p. Let F be the number of elements of ? whose orbit under this action has size 1. Prove

|?| ≡ F mod p.

(b) Let G be a group of order p^km and let P be a Sylow p-subgroup. Let Ω be the set of all Sylow p-subgroups of G, so |Ω| = n_p. Let P act on Ω by conjugation. That is, for any p ∈ P and Q ∈ Ω, p · Q = pQp^-1. Show that the only element of ? whose orbit under this action has size 1 is P, and use part (a) to deduce

n_p ≡ 1 mod p.

(c) With the same G and Ω as in part b), let G act on Ω by conjugation; denote this action by ·. Explain why the Sylow Theorem directly tells us that if P is a Sylow p-subgroup of G then |G · P| = |Ω|. Use this together with part (b) to deduce n_p|m.

3. Let p ≥ 2 be a prime. In this problem we will prove the following theorem:

"For any positive integer n ≥ 1, if G is a group of order pⁿ, then G has a subgroup of order p^k for every positive integer k < n."

When n = 1 there is no positive integer k < n so the base case is vacuously true, so we move on to the inductive step.

Inductive Step: Assume that the theorem is true for groups of order pⁿ. Now let G be a group of order pⁿ⁺¹.

(a) Using the Class Equation, prove |Z(G)| is a multiple of p, and deduce that G has a normal subgroup H with |H| = p.

(b) Use G/H to complete the inductive step of the proof.

4. (a) Find a Sylow 2-subgroup of S₄.

(b) Show that any group of order 28 is solvable.

(c) Let p, q,r be distinct primes with p, q, r dividing |G|. Prove that |G| ≥ n_p(p - 1) + n_q(q - 1) + n_r(r - 1) + 1, and conclude any group of order 105 has a normal subgroup. (Hint: Count elements in Sylow p-, q-, r-subgroups.)

5. Read the article "The Brains behind the Enigma Code Breaking before the Second World War" by Elisabeth Rakus-Andersson. Write a two page article summarizing the impact of group theory on society through this context. Your article should discuss some of the mathematics involved.

Article Link- https://www.math.hmc.edu/~omar/math171F14/enigma.pdf.