Midterm MSc. Exam
Department of MathematicsFerdowsiUniversity of Mashhad
27 November 2007
1. a. Define k-transitive and k-homogenous group .
b. Prove that if G is transitive and for every is (k-1)-transitive, then G is k-transitive.
c. Give an example of a Double transitive groups.
2. a. Define a non-trivial block and give an example.
b. Prove that if and are two blocks of , then is also a block of G.
c. is it true the union of two blocks are always a block?
3. a. Define a primitive and imprimitive group and give an example for each.
b. Let be a transitive group. Then prove that if for every is a maximal subgroup of G, then is primitive.
4. a. Define a transvection and prove that if is a linear functional and is a linear transformation by the rule is a transvection on a suitable hyperplan H and .
b. Prove that can be generated by transvections.
c. Compute the order of groups .