Syllabus

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Quiz 1 [soln]: Define a function that is one-to-one, but is

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.

$$\begin{array}{l||l|l|l} * &a&b&c \\ \hline\hline a&a&a&b \\ \hline b&b&b&b \\ \hline c&c&b&c \end{array}$$

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 $\{4, 6\}$.

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).

Exam 1 (20 September)

Exam 2 (18 October)

Exam 3 (15 November)