# Math 417 Homework 2: Due Friday January 31

Instructions: You can discuss these problems with others, but write up your solutions on your own (i.e., don't just copy someone else's solutions, else the feedback I give you won't help you much). Please be neat and write in full sentences.
• [1] Let n > 2. An element of h of Dn is nontrivial if it is not the identity. It is a rotation if (thinking of Dn as symmetries of a regular n-gon) it keeps the top side of the n-gon up. It is a reflection if it is nontrivial but some point on the perimeter of the n-gon is not moved. Determine whether the following statement is true or false. Justify it if it is true, or give a counterexample if it is false: Every nontrivial symmetry in Dn is either a reflection or a rotation.
• [2]
• (a) Prove that the product of two rotations in Dn is a rotation.
• (b) Prove that the product of two reflections in Dn is a rotation.
• [3] Consider an infinite strip of equally spaced H's in the plane:
... H H H H H H H H H H H H ...
Describe the symmetries of the strip. Is the group of symmetries abelian?
• [4] Use Euclid's algorithm (described in Chapter 0) to find a solution (m, n) of 13m + 8n = 1.