Name: Student number: METU MATH 116, Midterm 2 _{1 2 3 4 5 6 7 8 Σ}

Tuesday, May 3, 2011, at 17:40 (100 minutes), totally 60 points

Instructors: S.Finashin, S.Pamuk, D.Pierce, E.Solak Instructions: Please, be accurate and show clearly the logic of your solutions. Only the answers are not enough: indicate your calculations and arguments.

Problem 1. (9 pts) Of the set {1, 2, 3, 4, 5, 6, 7, 9}, let σ ∈ S_{9} be the permutation

[\begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 3 & 7 & 4 & 1 & 6 & 8 & 2 & 9 & 5 \end{matrix}] .

(a) Write σ as a product of disjoint cycles.

(b) Find σ^{100}. (Hint: find the order of σ).

(c) Determine whether permutation σ is even or odd.

Problem 2. (6 pts) Consider transpositions α = (12), β = (23), γ = (34) in S_{4}.

(a) Find the product αβγ.

(b) Find g ∈ S_{4} such that β = gαg^{−1}.

Problem 3. (8 pts)

(a) Find the number of cosets for subgroups 3ℤ ⊂ ℤ and ℤ ⊂ ℚ. In the first case, give an explicit list of these cosets.

(b) Find the group multiplication table of ℤ/3ℤ (pay attention that the group operation is addition).

(c) Prove that every element of ℚ/ℤ has finite order. Problem 4. (7 pts) Assume that G is a group of prime order p.

(a) What can be the order of an element g ∈ G ? (Explain !)

(b) Prove that G is a cyclic group.

Problem 5. (10 pts) Suppose that H and N are subgroups of a group G. Assume that subgroup N is normal, and let K = H ∩ N.

(a) Prove that K is a normal subgroup of H.

(b) Let A = {x ∈ G: xK = Kx}. Prove that H ⊆ A.

(c) Prove that A is a subgroup of G. Problem 6. (5 pts) Let H and K be normal subgroups of G such that G/H has order 5, and G/K has order 3. Prove that for any g ∈ G, g^{15} ∈ K ∩ H.

Problem 7. (6 pts) Consider subsets 4ℤ ⊂ ℤ, {0} ⊂ ℤ, ∅ ⊂ ℤ, ℤ ⊂ ℚ, (0, ∞) ⊂ ℝ, {[0], [1], [2], [4]} ⊂ ℤ_{6}, M_{2}(ℤ) ⊂ M_{2}(ℝ).

(a) Which of the above examples are NOT subrings? Why they are not subrings ?

(b) Which ones are NOT ideals? Why not ? Problem 8. (9 pts) In each of the following two subrings of ℤ_{12}, find a unity or show that there is no unity (multiplicative identity).

(a) 2ℤ_{12} = {[0], [2], [4], [6], [8], [10]}

(b) 3ℤ_{12} = {[0], [3], [6], [9]}

(c) Find all the zero divisors in the ring ℤ_{12}.