# 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 D
_{n} is *nontrivial* if it is not the identity.
It is a *rotation* if (thinking of D_{n} 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 D_{n} is either a reflection or a rotation.
- [2]
- (a) Prove that the product of two rotations in D
_{n} is a rotation.
- (b) Prove that the product of two reflections in D
_{n} 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.