305 – Homework IV

This homework set is due Wednesday, March 21, at the beginning of lecture.

1. In lecture we have used a few times the group $({\mathbb Z}/n{\mathbb Z})^*$ where $n>1$. Prove that this is in fact a group. Since associativity of multiplication is automatic, and clearly $1\in({\mathbb Z}/n{\mathbb Z})^*$, this requires two things: You must check that if $a$ and $b$ are relatively prime with $n$, then so is their product $ab$. Also, you need to check that if $a$ is relatively prime with $n$, then there is a $b$, also relatively prime with $n$, such that $ab\equiv 1\pmod n$.

[Here is a suggestion on how to approach this last part: Recall that $\gcd(\alpha,\beta)$ is a linear combination of $\alpha$ and $\beta$, that is, there are integers $x,y$ such that $\alpha x +\beta y=\gcd(\alpha,\beta)$.]

2. Suppose $(G,*)$ is a structure with the following properties: $*:G\times G\to G$; $*$ is associative; there is an identity $1\in G$ such that $x1=1x=x$ for any $x\in G$; and every element of $G$ has a right inverse, that is, for any $x\in G$ there is a $y\in G$ such that $xy=1$. Is $(G,*)$ a group? (If yes, provide a proof, if not, please exhibit a counterexample.)

3. For $x,y$ in the open interval $(0,1)$, define

$\displaystyle x\odot y=\frac{xy}{1-x-y+2xy}.$

Show that $((0,1),\odot)$ is an abelian group. What is the identity of this group? (This example comes from work of Marion Scheepers.)

4. Let $q>3$ be a prime number. Let $\infty$ be a new element, not in ${\mathbb Z}/q{\mathbb Z}$. Define the set $G(q)$ to consist of $\infty$ and of those elements of ${\mathbb Z}/q{\mathbb Z}$ whose square is not $-3$. For example,

$G(7)=\{0,1,3,4,6,\infty\}$.

Define an operation $\star$ on $G(q)$ by setting, for $x,y\in G(q)$,

$\displaystyle x\star y=\left\{\begin{array}{cl}x&\mbox{ if }y=\infty\\ y&\mbox{ if }x=\infty\\ \displaystyle \frac{xy-3}{x+y}&\mbox{ if }x\ne\infty\mbox{ and }y\ne \infty,\end{array}\right.$

where the computation of the fraction takes place in ${\mathbb Z}/q{\mathbb Z}$, and it is understood to be $\infty$ if the denominator but not the numerator vanishes.

Prove that $(G(q),\star)$ is an abelian group. (Note that part of what you need to show is that $x\star y$ is defined for all $x,y\in G(q)$ and always results in an element of $G(q)$.)

What is $4\star 1$ in $G(7)$?

Show that for any prime $q>3$, $1\in G(q)$ and ${\rm ord}_{G(q)}(1)=3$, and conclude that $3\mid |G(q)|$. Conclude that $-3$ is a square modulo $q$ iff $q\equiv 1\pmod 3$. (This last equivalence is an easy example of a deep result, the law of quadratic reciprocity in number theory.)

(This example comes from work of Paul Pollack.)

5. [Edited as the original version was nonsense.]

Suppose that $(D,*)$ is a group, and that the following conditions hold: There are two elements $s,t\in D$ different from the identity and from each other, and such that $s^2=t^2=1$ and $st$ has order 4. Suppose that any element of $D$ is a “word” in the letters $s,t$, that is, for any $g\in D$ there are $x_1,\dots,x_n$, with each $x_i$ equal to $s$ or $t$, and

$g=x_1\dots x_n,$

and suppose that any equality between two such words $x_1\dots x_n=y_1\dots y_m$ is a consequence of the three rules above, $s^2=t^2=(st)^4=1$. [This is an example of a presentation, that we will discuss later.]

Show that $D$ has size 8 and is isomorphic to the dihedral group $D_8$ of symmetries of the square. (Recall that this means that the multiplication tables of $D$ and $D_8$ coincide once we suitably identify their elements.)

6. This problem is extra credit. Recall that the modular group ${\mathcal M}$ consists of those functions $f$ of the form

$\displaystyle f(z)=\frac{az+b}{cz+d},$

where $ad-bc=1$, $a,b,c,d\in{\mathbb R}$, and $z\in{\mathbb C}\cup\{\infty\}$. Here, if $f(z)$ is a fraction where the denominator but not the numerator vanishes, we identify it with $\infty$, and we set $f(\infty)=a/c$ if $c\ne 0$ and $f(\infty)=\infty$ if $c=0$. The operation here is composition of functions.

Show that ${\mathcal M}$ is indeed a group. Show that if $f\in{\mathcal M}$, and $z$ is either $\infty$ or a complex number whose imaginary part is non-negative, then the same holds for $f(z)$. We denote by ${\mathbb H}$ this “upper half plane”,

${\mathbb H}=\{z\in{\mathbb C}\mid \mbox{the imaginary part of }z\mbox{ is }\ge0\}\cup\{\infty\}.$

Show that for any three distinct elements $a,b,c$ of ${\mathbb H}$ there is a unique $f\in{\mathcal M}$ such that $f(a)=0$, $f(b)=1$, $f(c)=\infty$.

Show that if $f\in{\mathcal M}$ and $l$ is a circle, or a line (including the point at infinity), then $f(l)$ is also a line or a circle.

These functions $f$ are examples of Möbius transformations. This video may help visualize some of what they do.