Reference no: EM132980354

Question 1

Let G = Z x {±1 }. Let : G X G → G be the binary operation defined by

(x₁, y₂) . (x₂, Y₂) = (x₁ + x₂y₁, Y₁ Y₂)

for all (x₁ , y₁ ), (x₂, y₂) ∈ G.

1. Prove that (G, .) is a group.

2. The subset H = {0} X (±1 ) of G is a subgroup of G. Prove that H is not a normal subgroup of G.

3. Is G isomorphic to the direct product group Z x (±1)?

4. Let N = Z X (1) . Prove that N is a normal subgroup of G and give an isomorphism from G/N to a familiar group that is not a quotient group.

Question 2

1. Is the permutation

in S₉, a product of 3-cycles?
2. Let α, β ∈ S₈ be the permutations α = (3, 2, 1, 7)(5, 6)(3, 6, 4, 8)(3, 8) ∈ S₈ and β = (1, 2)(3, 4)(5, 6)(7, 8). Is α¹⁰ conjugate to β in S₈? if the answer is yes, give a permutation σ ∈ S₈ such that σα¹⁰σ^-1 = If the answer is no, prove it.

Question 3

Let G be a group of order 20. Let H be a group of order 24. Prove that if Φ: G → H is a homomorphism, then 5 || ker Φ| and |H|  

Question 4

Recall that Z[x] denotes the set of polynomials in the variable x with integer coefficients. Let

L= ( f ∈ Z[x] : deg(f) ≤ 2).

Note that L is a free abelian group with respect to the operation of polynomial addition.

1. Consider the subset N of L consisting of all polynomials in L that have even coefficients and vanish at x = 1. Prove that N is a subgroup of L, explain why N is a free abelian group, and determine the rank of N.

2. Prove that L/N = Z x Z/2  Z x Z/2Z.

Question 5

1. Let G be a group of order 140 = 2² . 5 . 7. Use Sylow's theorems to prove that G has normal subgroup of order 5 and a normal subgroup of order 7.

2. Let n be a positive integer that is not divisible by 3 and let G be a group of order 3n. For that if x, y ∈ G are elements of order 3, then y is conjugate to either x or x^-1.