MAED 5121 Algebra and Its Applications I
Midterm Examination, Fall 2024
1. Let ~ be the relation defined on the set of integers Z by
n ~ m for n, m ∈ Z if n2 ≡ m2 mod 7;
i.e., n ~ m if n2 — m2 is a multiple of 7.
(a) Show that ~ is an equivalence relation. [8 points]
(b) How many elements (equivalence classes) are in the quotient set Z/~? Why? [8 points]
(c) Is the operation ⊞ on Z/~ defined by [x] ⊞ [y] = [x+y] for [x], [y] ∈ Z/~ a well-defined binary operation on Z/~? Why? [4 points]
(d) Is the operation ⊠ on Z/~ defined by [x] ⊠ [y] = [xy] for [x], [y] ∈ Z/~ a well-defined binary operation on Z/~? Why? [5 points]
2. Consider the cycles σ = (356) and τ = (1357) in the permutation group S7 .
(a) How many distinct 7-cycles does S7 have? [5 points]
(b) Express the permutation τ2 as a product of disjoint cycles. [5 points]
(c) Express the permutation σ —1τ as a product of disjoint cycles. [5 points]
(d) Find the order of the permutation (σ—1τ)360 . [5 points]
(e) Can you find a cycle P ∈ S7 such that P2 = (124)(356)? Show your work for full credit. [5 points]
3. Let G be the set of 3 × 3 invertible real matrices, which is a group under matrix multiplication. Let
(a) Show that is not an abelian group. [4 points]
(b) Prove that is a subgroup of G , and determine if H is abelian or not. [7 points]
(c) Show that = {A ∈ : AB = BA for all B ∈ } is a also a subgroup of . What kind of matrices are in ? [7 points]
(d) In the left coset decomposition of by the subgroup x , what are the elements in the left coset Cx ,
where
Is the left coset Cx a countable set, or uncountable set? [7 points]
4. Let f : G -→ H be a group homomorphism, and h : G -→ G is defined by h(x) = x2 . Prove the following:
(a) If G is abelian and f is surjective, then H is also abelian. [8 points]
(b) If f(G) has n distinct elements, then xn ∈ ker f for all x ∈ G. [8 points]
(c) G is abelian if and only if h is a group homomorphism. [9 points]