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MAT301 Assignment 4

Due:  23:59 EDT, Friday, July 14, 2023

Justify all claims in your solutions and state the results that you use. You may only use results that have been covered in lectures, tutorials, or in Chapters 0–7 or Chapters 9– 10 in Gallian.

Exercise 1.  Let p and q be distinct primes and let G be a group of order pk q for some positive integer k . Suppose that G has two distinct subgroups of order pk . Prove that p < q .

Exercise 2.  In this exercise, you will classify all non-abelian groups of order 8 up to isomorphism. Let G be a non-abelian group of order 8.

1. Prove that G has an element x of order 4.

2. Let y ∈ G \ ⟨x⟩ . Prove that G = {e,x,x2 ,x3 ,y,xy,x2 y,x3 y} .

3. Prove that either y2  = e or y2  = x2 , and either yx = x2 y or yx = x3 y .

4. Prove that if y2  = e, then yx = x3 y and G is isomorphic to the dihedral group D4  of order 8.

5. Prove that if y2  = x2 , then yx = x3 y and G is isomorphic to the dicyclic group Dic2  of order 8.  (The dicyclic group Dic2  of order 8 is equal to the quaternion group Q8 .)

Conclude that up to isomorphism the only non-abelian groups of order 8 are D4  and Dic2  = Q8 .

Exercise 3.  Let n ≥  1.  For each A ∈ GLn (R) and b ∈ Rn , define a map [A,b] : Rn  → Rn  by [A,b](x) = Ax + b for all x ∈ Rn .  Such transformations of Rn  are called invertible affine transformations of Rn .  Let Affn  = {[A,b] : A ∈ GLn (R),b ∈ Rn } .

1. Prove that Affn  is a group with respect to composition.

2. Prove that the subset T = {[In ,b] : b ∈ Rn } ⊂ Affn  is a normal subgroup Affn .

3. Describe the quotient group Affn /T.

Exercise 4. Let G be a group and let p be the least prime divisor of |G| . Using Theorem 7.2 in Gallian 9th ed., prove that any subgroup of index p in G is normal.

Exercise 5.  Prove that every non-trivial normal subgroup H of A5  contains a 3-cycle. (Hint: The 3-cycles are the non-identity elements of A5  with the largest number of fixed points.  If σ ∈  Sn , a reasonable way of trying to construct a permutation out of σ with more fixed points than σ  is to form a commutator [σ,τ] = στσ − 1 τ − 1  for an appropriate permutation τ ∈ Sn . This idea is used in the solution of Rubik’s cube. Why is this a reasonable thing to try?)