Exercise 5.1 Group
In Exercises 1 through 6, determine whether the binary operation ∗ gives a group structure on the given set. If no group results, give the first axiom in the order S1,S2,S3 from Definition that does not hold.
1. Let ∗ be defined on Z by letting a ∗ b = ab.
23. Let. Let ∗∗ be defined onbe defined on 2RZ+by letting= {2n | n ∈a ∗Zbb}==by lettingab√.ab . a ∗ b = a + b.
4. Let ∗ be defined on Q by letting a ∗
56. Let. Let ∗∗ be defined on the setbe defined on C by lettingR∗ of nonzero real numbers by lettinga ∗ b = |ab|. a ∗ b = a/b.
Exercise 5.2 Group
Let S be the set of all real numbers except −1. Define ∗ on S by
a ∗ b = a + b + ab.
a. Show that ∗ gives a binary operation on S.
b. Show that ⟨S,∗⟩ is a group. 2 ∗ x ∗ 3 = 7 in S.
c. Find the solution of the equation
Exercise 5.3 Subgroup
In Exercises 1 through 6, determine whether the given subset of the complex numbers is a subgroup of the group C of complex numbers under addition.
1. R
2. Q+
3. 7Z
4. The set iR of pure imaginary numbers including 0
5. The set πQ of rational multiples of π
6. The set {πn | n ∈ Z}
Exercise 5.4 Cyclic Group
In Exercises 1 through 5, find all orders of subgroups of the given group.
1. Z6
2. Z8
3. Z12 4. Z20
5. Z17
Exercise 5.5 Permutation Group
In Exercises 1 through 5 , compute the indicated product involving the following permutations in
Exercise 5.6 Permutation Group
In Exercises 1 through 4, compute the expressions shown for the permutations σ,τ and µ defined prior to Exercise 5.5 .
1.
2.
3.
4.
Exercise 5.7 Homomorphism
Determine whether the given map ϕ is a homomorphism.
1. Let ϕ : Z → R under addition be given by ϕ(n) = n.
2. Let ϕ : R → Z under addition be given by ϕ(x) = the greatest integer ≤ x.
3. Let ϕ : R∗ → R∗ under multiplication be given by ϕ(x) = |x|.
Exercise 5.8 Coset and Lagrange Theorem
1. Find all cosets of the subgroup 4Z of 2Z.
2. Find all cosets of the subgroup ⟨2⟩ of Z12.
3. Find all cosets of the subgroup ⟨4⟩ of Z12.
4. Find all cosets of the subgroup ⟨18⟩ of Z36.
5. Find the index of ⟨3⟩ in the group Z24.
6. Let σ = (1,2,5,4)(2,3) in S5. Find the index of ⟨σ⟩ in S5. 7. Let µ = (1,2,4,5)(3,6) in S6. Find the index of ⟨µ⟩ in S6.
