Math 407 Homework 1

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title: Homework 1
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### Directions
Solve the following problems and type up your solutions.  Your solutions should be provided in one of the following formats (in order of preference)
* typed up in $$\LaTeX$$ and submitted as a PDF on Canvas
* written legibly on blank paper, scanned into a PDF and then uploaded on Canvas
* written on ancient parchement with a quill and then flown to the instructor via owl post like in Harry Potter

If you go with the first strategy, you may wish to check out Overleaf which is a free and intuitive website for generating $$\LaTeX$$ documents online.
If you wish to use the second method and don't own a scanner at home, you can check out the numerous scanning apps available for smartphones.

**Problem 1:** The *hours on a clock* ($$1$$ through $$12$$) form a natural group with binary operation defined by our usual way of adding time.  For example if it's $$8$$ o'clock , then $$7$$ hours later it will be $$3$$ o'clock (ignoring AM and PM).  In this way $$7+8 = 3$$.  Likewise $$2+5 = 7$$, $$4+9=1$$ and so on.

* (A) What is the identity element of this group?
* (B) Give a formula for the inverse of an element $$k$$ of this group.
* (C) Is this group abelian or nonabelian?  Prove it.

**Problem 2:** Let $$\mathcal{S} = [-1,1]\times[-1,1]$$ be the unit square in $$\mathbb R^2$$.  A **symmetry of the square** is an invertible linear transformation $$T:\mathbb R^2\rightarrow\mathbb R^2$$ which maps vertices of $$\mathcal S$$ to vertices of $$\mathcal S$$.

* (A) Write down all the symmetries of the square using $$2\times 2$$ matrices.  How many are there?  
* (B) Write down a multiplication table for the set of matrices in (A).
* (C) Show that the set of matrices in (A) is a group.  Is this group abelian or nonabelian?  Prove it.

**Problem 3:** Suppose that $$G\subseteq \mathbb R^3$$ is a group with binary operation defined in terms of the cross product by

$$\vec u\ast\vec v = \vec u\times\vec v$$

* (A) Show that the identity element of $$G$$ must be $$\vec 0$$.
* (B) Use (A) to show that $$G$$ is the trivial group $$G = \{\vec 0\}$$.