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Finite Groups

Midterm MSc. Exam

Department of MathematicsFerdowsiUniversity of Mashhad

27 November 2007

A. Erfanian

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 .