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| = pkm with gcd(p, m) = 1. We will prove G has a subgroup of order pk.
(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| = pk}.
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 pk divides |H|.
(d) Show that |H| ≤ pk and hence by part (c), |H| = pk. Conclude the result.
2. (a) Let ? be any set and suppose G acts on ?. Further suppose |G| = pk 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 pkm and let P be a Sylow p-subgroup. Let Ω be the set of all Sylow p-subgroups of G, so |Ω| = np. 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
np ≡ 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 np|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 pk 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| ≥ np(p - 1) + nq(q - 1) + nr(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.