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∗bedefinedonZbylettinga∗b=ab.
2. Let∗bedefinedon2Z={2n|n∈Z}bylettinga∗b=a+b.
3. Let∗bedefinedonR+bylettinga∗b=√ab.
4. Let∗bedefinedonQbylettinga∗b=ab.
5. Let ∗ be defined on the set R∗ of nonzero real numbers by letting a ∗ b = a/b. 6. Let∗bedefinedonCbylettinga∗b=|ab|.
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.
c. Findthesolutionoftheequation2∗x∗3=7inS.
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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. Theset{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
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Exercise 5.5 Permutation Group
In Exercises 1 through 5 , compute the indicated product involving the following
permutationsinS6:σ=1 2 3 4 5 6, τ=1 2 3 4 5 6, μ=
123456
524316
1. τσ 2. τ2σ
3. μσ2
4. σ−2τ 5. σ−1τσ
314562 241365
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. |⟨σ⟩|
2. |⟨τ2⟩| 3. σ100 4. μ100
Exercise 5.6
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Exercise 5.7 Homomorphism
Determine whether the given map φ is a homomorphism.
1. Letφ:Z→Runderadditionbegivenbyφ(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 cosets of the subgroup 4Z of 2Z. cosets of the subgroup ⟨2⟩ of Z12. cosets of the subgroup ⟨4⟩ of Z12. cosets of the subgroup ⟨18⟩ of Z36.
1. Find all
2. Find all
3. Find all
4. Find all
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.
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Reference
1. Rosen, Kenneth H., and Kamala Krithivasan. Discrete mathematics and its applica- tions: with combinatorics and graph theory. Tata McGraw-Hill Education, 2012.
2. Fraleigh, John B. A first course in abstract algebra. Pearson Education India, 2003.
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