Cyclic Index of Automorphism Group

1. Find the cyclic index of automorphism group of the finite projective plane with 7 vertex. 2. Let  be a permutation on n objects. Define k   i (     id ). a. Let n a be the number of permutation on n objects with 3   i or 4   i . Find the function generator n n a x . b. Let n b be the number of permutation on n objects with k   i when k is odd. Find n n b x . c. By b, find n b . d. Find n b directly. 3. Prove: for all k r   0 there is 0 n k r ( , ) which for all 0 n n k r  ( , ) if [ ] n F k        is family of sets so for all S T F ,  : S T r   , then n r F 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 A B , ,  with same color ( A B A B B A    ( \ ) ( \ )