Reference no: EM132928849

Question 1: Let n ≥ 1. For each m ∈ { 1, ... , n} , define σ_m ∈ S_2n as follows. For each k ∈ { 1, ... , 2m}, define σ_m (k) = k+1/2 if k is odd, and σ_m(k) = m + k/2 if k is even.

1. Determine inv(σ_n), i.e. determine all inversions of σ_n, and prove a closed formula for |inv(σ_n)| .

2. For m ≥ 2, express σ_m as a product of an m-cycle and σ_m-1.

Question 2: Let G be a group. For each element a ∈ G with a² = e, define Φ_a : S_n -> G by Φ_a(σ) = a^1-sign(σ)/2 for all a ∈ Sn. That is, for all σ ∈ S_n we have that Φ_a (σ) is equal to e if σ is even and is equal to a if σ is odd.

1. Prove that Φ_a : S_n → G is a homomorphism for all every element a ∈ G with a² = e.

Let H be a group such that for all x, y ∈ H, if x² = e and y² = e then x and y commute.

2. Prove that if Φ : S_n -> H is a homomorphism, then 0 is constant on the set of transpositions in S_n.
3. Prove that {Φa : a ∈ G, a² = e} is the set of all homomorphisms from S_n to H.

A corollary of Part 3 is that for n > 1 there is a unique non-trivial homomorphism from S_n to {±1}, namely sgn.

Question 3:
Prove that A₁₅ is generated by the cycles
Π₁ = (5, 4, 3), Π₂ = (6, 5, 4, 3, 2),
Π₃ = (7, 6, 5, 4, 3, 2, 1),
σ₁ = (9, 8, 7), σ₂ = (10, 9, 8, 7, 6),
σ₃ = (11, 10, 9, 8, 7, 6, 5),

Τ₁ = (13, 12, 11), Τ₂₁ = (14, 13, 12, 11, 10),
Τ₃ = (15, 14, 13, 12, 11, 10, 9).