This homework set is due Wednesday, March 21, at the beginning of lecture.
1. In lecture we have used a few times the group where . Prove that this is in fact a group. Since associativity of multiplication is automatic, and clearly , this requires two things: You must check that if and are relatively prime with , then so is their product . Also, you need to check that if is relatively prime with , then there is a , also relatively prime with , such that .
[Here is a suggestion on how to approach this last part: Recall that is a linear combination of and , that is, there are integers such that .]
2. Suppose is a structure with the following properties: ; is associative; there is an identity such that for any ; and every element of has a right inverse, that is, for any there is a such that . Is a group? (If yes, provide a proof, if not, please exhibit a counterexample.)
3. For in the open interval , define
Show that is an abelian group. What is the identity of this group? (This example comes from work of Marion Scheepers.)
4. Let be a prime number. Let be a new element, not in . Define the set to consist of and of those elements of whose square is not . For example,
Define an operation on by setting, for ,
where the computation of the fraction takes place in , and it is understood to be if the denominator but not the numerator vanishes.
Prove that is an abelian group. (Note that part of what you need to show is that is defined for all and always results in an element of .)
What is in ?
Show that for any prime , and , and conclude that . Conclude that is a square modulo iff . (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 is a group, and that the following conditions hold: There are two elements different from the identity and from each other, and such that and has order 4. Suppose that any element of is a “word” in the letters , that is, for any there are , with each equal to or , and
and suppose that any equality between two such words is a consequence of the three rules above, . [This is an example of a presentation, that we will discuss later.]
Show that has size 8 and is isomorphic to the dihedral group of symmetries of the square. (Recall that this means that the multiplication tables of and coincide once we suitably identify their elements.)
6. This problem is extra credit. Recall that the modular group consists of those functions of the form
where , , and . Here, if is a fraction where the denominator but not the numerator vanishes, we identify it with , and we set if and if . The operation here is composition of functions.
Show that is indeed a group. Show that if , and is either or a complex number whose imaginary part is non-negative, then the same holds for . We denote by this “upper half plane”,
Show that for any three distinct elements of there is a unique such that , , .
Show that if and is a circle, or a line (including the point at infinity), then is also a line or a circle.