MAED 5121 Algebra and Its Applications I
Midterm Examination, Fall 2022
1. Let f : U → W is a mapping from the set U to the set W, and let X1, X2, . . . , Xn be subsets U.
(a) Prove that f [10 points]
(b) Let A1, A2, . . . , An be the sets defined by for all integers 2 ≤ k ≤ n. Prove that Ai ∩ Aj = ∅ for all i ≠ j, and n[i=1Ai =n[i=1Xi
. [15 points]
2. Let G = {(a, b) ∈ R × R : a ≠ 0}, and let ∗ be the operation on G defined by (a, b) ∗ (c, d) = (ac, bc + d) via the addition and multiplication of real numbers.
(a) Show that G is a non-abelian group under the operation ∗. [10 points]
(b) Show that H = {(a, 0) ∈ R × R : a ≠ 0} is a subgroup of G. [5 points]
(c) What are the elements in a left coset (c, d)H of the subgroup H? [5 points]
(d) Let G/H = {(c, d)H : c, d ∈ R, c ≠ 0} be the set of all left cosets of H. Find a bijection from G/H to the interval [0, ∞). [5 points]
3. Consider the cycles σ = (3517) and τ = (123) in the permutation group S7.
(a) Express the permutation σ−1τ as a product of disjoint cycles. [6 points]
(b) Find the order of σ−1τ. [4 points]
(c) Prove that for any permutation
α(3517)α−1 is also a 4-cycle. [10 points]
(Hint: Determine how the product permutation maps α(1), α(2), . . . , α(7).)
(d) Find a permutation β ∈ S7 such that β(3517)β−1 = (1234). [5 points]
4. Let G be a group. Define a relation ∼ on G by a ∼ b if ab−1x = xab−1 for all x ∈ G.
(a) Prove that ∼ is an equivalence relation. [10 points]
(b) If G = S3 is the permutation group with six permutations, find all elements in the equivalence class [(123)] of the permutation (123). [7 points]
(c) How many distinct elements are in the quotient set S3/∼? (Show your reasoning for full credit.) [8 points]