Syllabus
Homework
For Exam 1
HW1 (due 30 Aug) [soln]: p.8 #1,2,6,7,8,9,16,24,25,30,31,35; p.25 #1,2,3,4,5,6,9,10
(for graduate students): p.8 #18,28,32,33,36; p.25 #11,12
HW2 (due 6 Sep) [soln]: p.8 #24,25,30,31,35; p.25 #1,2,3,4,5,6,9,10,18,19; p.34 #4,5,6,7,11,12
(for graduate students): p.8 #28,32,33,36; p.25 #11,12; p.34 #13,14,15
HW3 (due 13 Sep) [soln]: p.34 #4,5,6,7,11,12,13,14
(for graduate students): p.34 #8,9,10,14,15
For Exam 2
HW4 (due 25 Sep) [soln]: p.36 #26, 27; p.45 #1, 2, 4, 5, 6, 12, 13, 14, 15, 19, 33, 36
(for graduate students): p.36 #28; p.45 #10, 24, 31, 35
HW5 (due 29 Sep) [soln]: p.55 #4, 5, 6, 7, 8, 9, 11, 12, 23, 24, 27, 28, 41, 43, 51, 54
(for graduate students): #46, 47, 49
HW6 (due 4 Oct) [soln]: p.55 #21, 22, 25, 33, 34, 35; p.66 #1, 2, 3, 4
(for graduate students): none!
For Exam 3
HW7 (due 23 Oct) [soln]: p.66: #5, 6, 7, 8, 9, 10, 17, 18, 22, 23, 24, 25, 26, 44
(for graduate students): #46, 48
HW8 (due 23 Oct) [soln]: p.72: #1, 2, 5, 6, 15, 16; p.83: #1, 2, 3, 4, 5, 18a, 30, 31
(for graduate students): p.83: #20, 46
HW9 (due 1 Nov) [soln]: p.83: #11, 12, 13; p.94: #1, 2, 3, 7, 8, 10, 11, 30, 31; p.100 #1, 2, 3, 4
(for graduate students): p.96: #36, 37
HW10 (due 8 Nov) [soln]: p.101: #6, 9, 28, 34; p.110: #1, 3, 4, 9, 10, 15, 16, 46, 54; p.133: #1, 2, 3, 4, 9, 10
(for graduate students): p.101: #41, 43; p.112: #39, 41
For Final Exam
HW11 (due 27 Nov29 Nov): p. 133 #1, 2, 3, 4, 10, 11, 12, 13, 17, 18, 19, 44, 49; p. 142: #1, 2, 3, 4, 9, 10, 11, 12, 13, 14, 31, 35
(for graduate students): p.135 #47, 50; p.143 #33, 40
HW12 (due 8 Dec): p. 142: #1, 2, 3, 4, 9, 10, 12, 13, 14, 31, 35
(for graduatae students): none
Quizzes
For Exam 1
Quiz 1 [soln]: Define a function that is one-to-one, but is not onto.
Quiz 2 (due 8 Sep) [soln]: Complete the following table so that it makes $*$ into a commutative binary operation:
$$\begin{array}{l|l|l|l}
*&a&b&c \\
\hline
a&b&&c \\
\hline
b&&b& \\
\hline
c&&&b
\end{array}$$
Quiz 3 (due 14 Sep) [soln]: Let $*$ be defined on $\mathbb{R}^+=(0,\infty)$ (positive real numbers) by letting $a*b=\sqrt{ab}$. Does the structure $\left\langle \mathbb{R}^+, * \right\rangle$ obey all of the group axioms $\mathscr{G}_1$, $\mathscr{G}_2$, and $\mathscr{G}_3$? If not, then which ones does it not satisfy?
Quiz 4 (due 19 Sep) [soln]: Does the binary operation $*$ defined in the following table obey the group axioms $\mathscr{G}_2$ and $\mathscr{G}_3$ of a group on the set $G=\{a,b,c\}$? Is it a commutative operation? Explain.
$$\begin{array}{l||l|l|l}
* &a&b&c \\
\hline\hline
a&a&a&b \\
\hline
b&b&b&b \\
\hline
c&c&b&c
\end{array}$$
For Exam 2
Quiz 5 (due 26 Sep) [soln]: Argue why $\pi\mathbb{Z}=\left\{\pi n \colon n \in \mathbb{Z}\right\}$ is a subgroup of $\left\langle \mathbb{R},+\right\rangle$.
Quiz 6 (due 13 Oct) [soln]: Consider a set $G=\{a,b,c,d\}$ and a binary operation $*$ given by the following table:
$$\begin{array}{l|l|l|l}
* & a & b & c & d \\
\hline
a & a & b & c & d \\
\hline
b & b & c & d & a\\
\hline
c & c & d & a & b \\
\hline
d & d & a & b & c
\end{array}$$
First, find the groups generated by each of the four elements of $G$ - you should find that $G$ is cyclic -- what are its generators? By a theorem proved in lecture, we know that $\langle G,*\rangle$ is isomorphic to $\mathbb{Z}_4$. Since this is the case, please provide an isomorphism between $\langle G,*\rangle$ and $\langle \mathbb{Z}_4,+\text{ mod } 4\rangle$.
Quiz 7 (due 16 Oct) [soln]: Consider the group $\left\langle \mathbb{Z}_{10},+\text{ mod }10\right\rangle$. Find all subgroups of the group and draw its subgroup diagram.
Quiz 8 (due 16 Oct) [soln]: Consider the group $\langle \mathbb{Z}_{14}, +\text{ mod }14\rangle$. Findthe group generated by the generating set $\{4, 6\}$.
For Exam 3
Quiz 9 (due 20 Oct) [soln]: Consider the symmetric group $\langle S_4, \circ\rangle$ and let $\sigma=\begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 3 & 1 \end{pmatrix}$. Find the cyclic group generated by $\sigma$, i.e. $\langle \sigma \rangle$.
Quiz 10 (due 24 Oct) [soln]: Write the permutation $\sigma=\begin{pmatrix} 1&2&3&4&5&6&7&8 \\ 5&7&2&6&1&3&8&4 \end{pmatrix}$ as a product of disjoint cycles AND as a product of transpositions.
Quiz 11 (due 31 Oct) [soln]: Verify Theorem 10.14 as was done in the 30 October class when $G=\mathbb{Z}_{18}$, $H=\langle 3 \rangle$, and $K=\langle 6 \rangle$ (notice $K$ is a subgroup of $H$).
Quiz 12 (due 2 Nov) [soln]: Find subgroups generated by elements of $\mathbb{Z}_3 \times \mathbb{Z}_4$ to determine if it is cyclic (and hence also isomorphic to $\mathbb{Z}_{12}$) or not cyclic.
Quiz 13 (due 7 Nov) [soln]: Find all abelian groups of order 120 (up to isomorphism).
Exams
Exam 1 (20 September)
Exam 2 (18 October)
Exam 3 (15 November)
