**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 ( \ ) ( \ )