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{\bf Math 113 Homework 5, due 2/26/2019}
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{\bf \noindent\large Make sure you are using the 7th edition of Abstract algebra by Fraleigh -- if you do the wrong problems, you won't get points!}

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{\bf 1.} Say $(G, *)$ is a group and $H\subseteq (G,*)$ is a subset that is closed under the operation $*$. Then whether or not $H$ is a subgroup, we can see that $(H, *)$ (operation inherited from $G$) is a valid binary structure. Show that if this binary structure is a group, then $H$ is a group in the sense described in class, namely: $e_G\in H$ and if $h\in H$ then $h^{-1_G}\in H$ (you don't have to check closure, since this is one of our assumptions on the subset $H$). Hint: the group property of $H$ implies there is an identity element $a\in H$. What is $a*a$? Remember that the operation $a*a$ doesn't depend on whether $a$ is understood as an element of $H$ or of $G$.

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{\noindent \bf (2.-11.) Define $U\subseteq (\C^*, \cdot)$ to be the ``unit circle'', i.e.\ the subset $z\in \C^*\mid |z| = 1.$ The following problems will have to do with symmetries of the circle $U$ given by rotations and reflections. Don't worry too much here about set-theoretic details and rigor: the idea is to do some computations and get a geometric picture of what are the ``symmetry'' functions $f:U\to U$ and how they compose.}

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{\bf 2.} Show that under multiplication, $U$ is a subgroup of the multiplicative group $\C^*, \cdot$. 

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{\bf 3.} Define the function $e: \R\to U$ with $e(r) : = \cos(r) + i\cdot \sin(r).$ Show that $e$ does in fact take values in $U$, and check that $e$ satisfies the homomorphism property (where $\R$ is viewed as a group with additive structure).

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{\bf 4.} We say $r \equiv_{2\pi} s$ if $r-s$ is an integer multiple of $2\pi.$ Show that $\equiv_{2\pi}$ is an equivalence relation and that $r \sim r', s \sim s'\implies r+s\sim r+s',$ so the addition operation $[r] + [s] : = [r+s]$ is well-defined. Once well-definedness is checked, the group properties for $\R$ imply that $\R/\equiv_{2\pi}$ with the addition operation on classes given above is a group (once you check well-definedness you automatically get that it is a group with identity $[0]$ and $[a]^{-1} = [-a]$, no need to prove this).

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{\noindent \bf For future problems, write $\text{Rad}: = \R/\equiv_{2\pi}$ ``the group of radians'': elements represent angles in radian notation, so that for example $[\pi] = [-\pi] = [3\pi]$ corresponds to the angle $180^\circ$ and $[\pi/2] = [-3\pi/2]$ is $90^\circ$. The operation $+$ defined in problem 4 is ``angle addition''.}

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{\bf 5.} Let's define the function $\phi:\text{Rad}\to U$ (where $\text{Rad} : = \R/\equiv_{2\pi}$) by $\phi([r]) : = e(r).$ (The symbol $\phi$ is the greek letter Phi.) Show that $\phi$ is well-defined. Show that it is an isomorphism (hint: you may use the homomorphism property of the function $e$.)

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      {\bf 6.} For $\theta\in \text{Rad}$ (defined in problem 4), define the function $\r_\theta: U\to U$ by $\r_\theta(z) : = \phi(\theta)\cdot z$. Check that indeed, $\r_\theta(z)\in U$ if $z\in U.$ Check that, expressing $z = x + y\cdot i,$ the operation $\r_\theta$ rotates the point $(x,y)$ by $\theta$ degrees counterclockwise around the origin, i.e.\ applies the matrix $$\begin{pmatrix}\cos(\theta)& -\sin(\theta)\\ \sin(\theta)&\cos(\theta) \end{pmatrix}$$ to the vector $\begin{pmatrix}x\\ y\end{pmatrix}.$

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{\bf 7.} Show that the composition $\r_{[\theta]}\circ \r_{[\theta']} = \r_{[\theta + \theta']},$ where all are interpreted as functions from $U$ to itself. (Remember that two functions are the same if all their values are the same and don't try to be fancy or use matrices here.)

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{\bf 8.} Define the function $s_\theta:U\to U$ by $s_\theta(u) : = \frac{\phi(\theta)}{u}$ (here ``$s$'' stands for symmetry: this is a reflection function). Check that $s_\theta(u)\in U$ if $u\in U$, so $s_\theta$ makes sense as a function from $U$ to itself. %This is the function that takes a point of the circle $U\subseteq \C$ and \emph{reflects} it across the line that meets the circle $U$ at the opposite angles $\phi[\theta/2]$ and $\phi[-\theta/2]$.

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{\bf 9.} As an example, draw the four points $1, i, -1, -i$ in $U$ (corresponding to angles $[0], [\pi/2], [\pi], [3\pi/2]$). Draw an arrow from each of these points $z$ to $\r_{\pi/2}(z)$ (no proof needed: note that $\r_{\pi/2}(z)$ should once again be one of these four points). 

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Repeat for $\r_{\pi}, s_{\pi/2},$ and $s_\pi$. Notice that $s_\theta$ is always a \emph{reflection} function (no proofs needed).

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{\bf 10.} Here is how you can compute $s_\theta\circ s_{\theta'}$ (composition for two different angles): $$s_\theta\circ s_{\theta'}(u) = \frac{\phi(\theta)}{\left(\frac{\phi(\theta')}{u}\right)} = u\cdot \frac{\phi(\theta)}{\phi(\theta')} = u\cdot \phi(\theta-\theta') = \r_\theta(u).$$ This shows that $s_\theta\circ s_{\theta'} = \r_{\theta-\theta'}.$ Compute using a similar argument the compositions $\r_\theta\circ s_{\theta'}$ and $s_{\theta'}\circ \r_\theta$ (warning: not abelian!). Together with the calculation for $\r_\theta\circ \r_{\theta'}$ done above, deduce that composing different functions of the form $\r_\theta$ or $s_\theta$ once again produces functions either of the form $\r_\theta$ or $s_\theta.$ In other words, the combined set $$Sym_U : = \{\r_\theta\mid \theta\in \text{Rad}\}\cup \{s_\theta\mid \theta\in \text{Rad}\}$$ of functions from $U$ to itself is closed under composition. The set of functions $Sym_{U}$ is the ``set of (distance-preserving) symmetries of the circle'' $U\subseteq \C$ (any symmetry of a circle that doesn't change distances between points is a rotation or a reflection). This group is also called $O(2)$, the group of orthogonal transformations of a two-dimensional vector space.

{\bf 11.} Show that the set of functions $Sym_U$ with operation $\circ$ (composition) is a group. You may assume composition of functions is associative (so you do not need to prove the assosciativity axiom). Note: don't waste time showing these symmetry functions are bijective (and therefore invertible): the composition rules you've found above will let you quickly find the inverse. Make sure to check, however, that the inverse you're defining is two-sided!

{\bf 12. Extra credit:} An element of a group $g\in (G, \cdot)$ is ``central'' if $g\cdot x = x\cdot g$ for any other $x\in G$. Show that $Sym_U$ has exactly two central elements, and they form a subgroup.


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