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PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

Thomas W. Judson, Isaac Newton (Editor)

Appendix B Hints and Answers to Selected Odd Exercises

2 The Integers
2.4 Exercises

2.4.1.

Answer.
The base case, \(S(1): [1(1 + 1)(2(1) + 1)]/6 = 1 = 1^2\) is true.
Assume that \(S(k): 1^2 + 2^2 + \cdots + k^2 = [k(k + 1)(2k + 1)]/6\) is true. Then
\begin{align*} 1^2 + 2^2 + \cdots + k^2 + (k + 1)^2 & = [k(k + 1)(2k + 1)]/6 + (k + 1)^2\\ & = [(k + 1)((k + 1) + 1)(2(k + 1) + 1)]/6, \end{align*}
and so \(S(k + 1)\) is true. Thus, \(S(n)\) is true for all positive integers \(n\text{.}\)

2.4.3.

Answer.
The base case, \(S(4): 4! = 24 \gt 16 =2^4\) is true. Assume \(S(k): k! \gt 2^k\) is true. Then \((k + 1)! = k! (k + 1) \gt 2^k \cdot 2 = 2^{k + 1}\text{,}\) so \(S(k + 1)\) is true. Thus, \(S(n)\) is true for all positive integers \(n\text{.}\)

2.4.11.

Hint.
The base case, \(S(0): (1 + x)^0 - 1 = 0 \geq 0 = 0 \cdot x\) is true. Assume \(S(k): (1 + x)^k -1 \geq kx\) is true. Then
\begin{align*} (1 + x)^{k + 1} - 1 & = (1 + x)(1 + x)^k -1\\ & = (1 + x)^k + x(1 + x)^k - 1\\ & \geq kx + x(1 + x)^k\\ & \geq kx + x\\ & = (k + 1)x, \end{align*}
so \(S(k + 1)\) is true. Therefore, \(S(n)\) is true for all positive integers \(n\text{.}\)

2.4.19.

Hint.
Use the Fundamental Theorem of Arithmetic.

2.4.23.

Hint.
Let \(S = \{s \in {\mathbb N} : a \mid s\text{,}\) \(b \mid s \}\text{.}\) Then \(S \neq \emptyset\text{,}\) since \(|ab| \in S\text{.}\) By the Principle of Well-Ordering, \(S\) contains a least element \(m\text{.}\) To show uniqueness, suppose that \(a \mid n\) and \(b \mid n\) for some \(n \in {\mathbb N}\text{.}\) By the division algorithm, there exist unique integers \(q\) and \(r\) such that \(n = mq + r\text{,}\) where \(0 \leq r \lt m\text{.}\) Since \(a\) and \(b\) divide both \(m\text{,}\) and \(n\text{,}\) it must be the case that \(a\) and \(b\) both divide \(r\text{.}\) Thus, \(r = 0\) by the minimality of \(m\text{.}\) Therefore, \(m \mid n\text{.}\)

2.4.27.

Hint.
Since \(\gcd(a,b) = 1\text{,}\) there exist integers \(r\) and \(s\) such that \(ar + bs = 1\text{.}\) Thus, \(acr + bcs = c\text{.}\) Since \(a\) divides both \(bc\) and itself, \(a\) must divide \(c\text{.}\)

2.4.29.

Hint.
Every prime must be of the form 2, 3, \(6n + 1\text{,}\) or \(6n + 5\text{.}\) Suppose there are only finitely many primes of the form \(6k + 5\text{.}\)

3 Groups
3.5 Exercises

3.5.1.

Hint.
(a) \(3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}\text{;}\) (c) \(18 + 26 \mathbb Z\text{;}\) (e) \(5 + 6 \mathbb Z\text{.}\)

3.5.15.

Hint.
There is a nonabelian group containing six elements.

3.5.17.

Hint.
The are five different groups of order 8.

3.5.25.

Hint.
\begin{align*} (aba^{-1})^n & = (aba^{-1})(aba^{-1}) \cdots (aba^{-1})\\ & = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1}\\ & = ab^na^{-1}. \end{align*}

3.5.31.

Hint.
Since \(abab = (ab)^2 = e = a^2 b^2 = aabb\text{,}\) we know that \(ba = ab\text{.}\)

3.5.35.

Hint.
\(H_1 = \{ id \}\text{,}\) \(H_2 = \{ id, \rho_1, \rho_2 \}\text{,}\) \(H_3 = \{ id, \mu_1 \}\text{,}\) \(H_4 = \{ id, \mu_2 \}\text{,}\) \(H_5 = \{ id, \mu_3 \}\text{,}\) \(S_3\text{.}\)

3.5.41.

Hint.
The identity of \(G\) is \(1 = 1 + 0 \sqrt{2}\text{.}\) Since \((a + b \sqrt{2}\, )(c + d \sqrt{2}\, ) = (ac + 2bd) + (ad + bc)\sqrt{2}\text{,}\) \(G\) is closed under multiplication. Finally, \((a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)\text{.}\)

3.5.49.

Hint.
Since \(a^4b = ba\text{,}\) it must be the case that \(b = a^6 b = a^2 b a\text{,}\) and we can conclude that \(ab = a^3 b a = ba\text{.}\)

3.5.55.

Answer.
\(1\)

3.5.57.

Answer.
\(n\)

3.5.59.

3.5.59.a
Answer.
\(2\)
3.5.59.b
3.5.59.b.i
Answer.
\(6\)
3.5.59.b.ii
Answer.
\(10\)

5 Runestone Testing
5.8 True/False Exercises

5.8.1. True/False.

Hint.
\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)

5.9 Multiple Choice Exercises

5.9.1. Multiple-Choice, Not Randomized, One Answer.

Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?

5.9.3. Multiple-Choice, Not Randomized, Multiple Answers.

Hint.
Do you know the acronym…ROY G BIV for the colors of a rainbow, and their order?

5.9.5. Multiple-Choice, Randomized, Multiple Answers.

Hint.
Do you know the acronym…ROY G BIV for the colors of a rainbow, and their order?

5.9.7. Multiple-Choice, Not Randomized, One Answer.

Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?

5.10 Parsons Exercises

5.10.1. Parsons Problem, Mathematical Proof.

Hint.
Dorothy will not be much help with this proof.

5.12 Matching Exercises

5.12.3. Matching Problem, Linear Algebra.

Hint.
For openers, a basis for a subspace must be a subset of the subspace.

5.13 Clickable Area Exercises

5.13.3. Clickable Areas, Text in a Table.

Hint.
Python boolean variables begin with capital latters.

5.18 Fill-In Exercises

5.18.11. Fill-In, Dynamic Math with Formulas as Answers.

5.20 Exercises that are Timed

Timed Exercises

5.20.1. True/False.
Hint.
\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)

5.27 Group Exercises

5.27.1. Multiple-Choice, Not Randomized, One Answer.

Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?
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