Mathematics 243, section 1 – Algebraic Structures Exam 3 – December 1, 2006

Directions Do all work in the blue exam booklet. There are 100 possible points.

I. In an RSA public key cryptosystem, the public key information is m = 323 and e = 11. Messages consisting of capital roman letters and blanks are encoded as 3-digit blocks 000, 001, ⋯, 026 (with A = 000, B = 001, ... , Z = 025, and blank = 026) and encrypted as 3-digit blocks. A) (15) How would the letters US be encrypted? B) (10) What is the (secret) decryption function g? II. (20) Let

H = \{A = ( \begin{matrix}a & 0 \\ b & c \end{matrix} ) : a, b, c ∈ R and a, c ≠ 0\}

Is H a group under the operation of matrix multiplication? If so, give a proof. If not, say which of the group properties fail. Note: We discussed a general rule for finding the inverse of a 2 × 2 matrix in class. If you don’t remember it, you can “buy” it from me during the exam in return for a reduction of 5 points on your score for this problem. III. Let G be a group, let a ∈ G be a fixed element, and consider the mapping ℓ_{a} : G → G defined by ℓ_{a}(x) = ax for all x ∈ G. A) (10) Show that ℓ_{a} is one-to-one and onto. B) (5) What is the inverse mapping of ℓ_{a}? IV. A) (15) Let G be a cyclic group with generator a. Show that every subgroup of G is also

cyclic. B) (10) G = Z_{30} is a group under the operation of addition mod 30. Find all of the

generators of the cyclic subgroup H = ⟨[18]⟩. V. Let G, H be two groups. A) (5) What is the definition of a group homomorphism from G to H? B) (10) Let G = Z_{24} with operation addition mod 24. What is the kernel of the group

homomorphism ϕ : G → G defined by ϕ([x]) = [9x]?

The General Pattern for 2 × 2 Matrix Inverses If

A = ( \begin{matrix}a & b \\ c & d \end{matrix} )

then A has a inverse matrix (for multiplication) if and only if det(A) = ad − bc ≠ 0, and then

A^{−1} = ( \begin{matrix}\frac{d}{ad−bc} & \frac{−b}{ad−bc} \\ \frac{−c}{ad−bc} & \frac{a}{ad−bc} \end{matrix} )