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.

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