Find the cyclic index of automorphism group of the finite

Question

1. Find the cyclic index of automorphism group of the finite projective plane with 7 vertex.

2. Let Π be a permutation on n objects. Define Π^k = i (Π o Π ooo Π =id).

a. Let a_n be the number of permutation on n objects with Π³ = i or Π⁴ = i .

Find the function generator ∑a_nXⁿ

b. Let b_n be the number of permutation on n objects with Π^k = i when k is odd. Find ∑b_nXⁿ

c. By b, find b_n.

d. Find b_n directly.

3. Prove: for all k > r ≥ 0 there is n₀ (k, r) which for all n > n₀(k, r) if F ⊂ ([n])/k) is family of sets so for all S ,T ∈ F : |S ∩ T| ≥ r, then |F| ≤ (n-r/k-r) (hint: first prove that this is saved under shifting).

4. Find integer n so that for all counting on the cube 2^[n] with seven colors:

a. There are 3 different sets A, B, A ∩ B with same color.

b. There are 3 different sets A, B, AΔB with same color ( AΔB = (A\ B) ∪ (B \ A)).