I need help on my project .(presentation)

Math 5420 Project ideas 1. Quaternion group . We know that we can expand the real line by de ning the square root of -1, and there is a clear relationship between the complex numbers Cand the 2-dimensional plane R2 . Can we develop an analogous result for higher dimensions { that is to say, can we further expand C? In 1843, William Hamilton answered this question by developing a system of quadruples ( a; b; c; d) that satis ed rules of addition, scalar multiplication, and quaternion multiplication. Show what these operations are, de ne the structure they create, and show how they can be de ned using matrices (example 3.3.7 p. 122 in Beachy and Blair). You might include a table for the quaterion group Q 8, some historical notes about Hamilton, and some words about the applications of quaternions in physics, computer graphics, etc.

2. Rubik's cube group . Discuss solving a Rubik's cube in terms of group theory. Can you describe the Rubik's cube group in terms of simpler, smaller groups?

3. Cryptography . Explain the di erence between private-key and public-key cryptography, show how the RSA cryptosystem works and how it is related to factoring of large numbers, and do some examples. A good resource is chapter 7 of the book Abstract Algebra, Theory and Applications by Thomas W. Judson, available free online.

4. Wallpaper groups . Wallpaper groups are two-dimensional symmetry groups. They catego- rize patterns by their symmetries, and can be described as isometries of the Euclidean plane that contain two linearly independent translations. There are 17 possible wallpaper groups.

Describe how these are de ned, give examples, and describe some properties. A good resource is chapter 12 of the book Abstract Algebra, Theory and Applications by Thomas W. Judson.

5. Lorentz group . Start with Problem 18, p. 124 of Beachy and Blair which de nes the Lorentz group and asks you to prove it is a subgroup of GL 2( R ). Then explain how this is related to Special Relativity. You might include an explanation of the di erences between Newtonian dynamics and Special Relativity, and some history about the development of Special Relativity.

6. Orbit-Stabilizer Theorem . Problems 19-22 on page 114 of Beachy and Blair de ne the sets C(a ) and Z(G ) and give some examples using S 3 and GL 2( R ). You have done 19 and 21, so do the others. In addition to nding the centralizer of (1 ;2 ;3) in S 3, also nd that set in S 4, and in A 4. Centers and centralizers are related to group actions. De ne what a group action is and and give some examples. You can use Beachy and Blair or Judson as a reference.

An important and fairly accesible result in this area is the Orbit-Stabilizer Theorem, state this theorem and give an explanation of the proof.

If there are other topics you would prefer to do, please see me or send me an email.