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\def\R{\mathbb R}
\def\Z{\mathbb Z}
\def\CP{\mathbb {CP}}
\def\H{\mathbb H}
\def\E{\mathbb E}
\def\S{\mathbb S}
\def\F{\mathcal F}
\def\D{\mathcal D}
\def\C{\mathbb C}
\def\Q{\mathbb Q}
\def\A{\mathbb A}
\def\til{\widetilde}
\def\M{\mathcal M}
\def\O{\mathcal O}
\def\N{\mathbb N}
\def\RP{\mathbb{RP}}
\def\T{\mathcal T}
\def\U{\mathcal U}
\def\V{\mathcal V}
\def\G{\mathcal G}
\def\Aut{\text{Aut}}
\def\Inn{\text{Inn}}
\def\Out{\text{Out}}
\def\Isom{\text{Isom}}
\def\til{\widetilde}
\def\1{{\bf 1}}
\def\id{\text{id}}

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\title{Homework 1 --- Groups}

\author{Danny Calegari}
\address{Department of Mathematics \\ Harvard \\ Cambridge, MA 02138}
\email{dannyc@math.harvard.edu}

\maketitle

Homework is assigned on Fridays; it is due at the start of class the week
after it is assigned. So this homework is due September 29th. Problems
marked ``Hard'' are extra credit; in other words, doing all the ordinary
problems correctly will earn full credit. But by doing hard problems, it
is possible to make mistakes on, or fail to complete, all the ordinary
problems, and still earn full credit.

\begin{prob}
Verify that each of the following examples are groups under the proposed
operation, and calculate the identity and the inverse of a general element.
\begin{enumerate}
\item{The integers $\Z$ under $+$}
\item{The set of $2\times 2$ matrices with real entries and nonzero determinant under
matrix multiplication. 
This is called the {\em general linear group in dimension $2$} and denoted $GL_2(\R)$}
\item{The set of $2\times 2$ matrices of the form $\left[\begin{matrix}
a & b \\ 0 & 1 \\ \end{matrix}\right]$ where $a$ is nonzero and $b$ is arbitrary under
matrix multiplication.}
\item{The group of {\em integers modulo $n$} under $+$. These are the sets
of equivalence classes of integers, where $a \sim b$ if and only if $a-b$ is
divisible by $n$. This group is denoted $\Z/n\Z$.}
\item{The set of $2\times 2$ matrices with entries in $\Z/2\Z$ (i.e. ``even'' and
``odd'') with the usual rules of multiplication and addition for
even and odd numbers, with odd determinant, under matrix multiplication. This
group is denoted $GL_2(\Z/2\Z)$.}
\item{The group of permutations of $n$ objects (which might as well be the
set $\lbrace 1, \dots , n \rbrace$) under composition. This group is called the
{\em symmetric group $S_n$}.}
\item{The group of symmetries of a regular $n$--gon under composition.
This group is called the {\em dihedral group $D_n$}.}
\item{For an arbitrary set $X$, the group of $1$--$1$ and invertible maps $X \to X$
under composition is a group. This group is called the {\em symmetric group of
$X$} and denoted $S_X$.}
\end{enumerate}
\end{prob}

\begin{prob}
How many elements are in $D_n$? How many are in $S_n$?
\end{prob}

\begin{prob}
List all the homomorphisms from $\Z,+$ to $\Z,+$. List all the homomorphisms
from $\Z/2\Z,+$ to $\Z,+$. 
\end{prob}

\begin{prob}
Let $S$ be a set and $G$ be a group. Show that the set of maps $f:S \to G$ is
a group $\text{Map}(S,G)$, 
under the operation $c(f,g)(s) = f(s)g(s)$ where the product on the
right hand side is taken in the group $G$. Suppose $S$ has two elements.
Show that $\text{Map}(S,G) \cong G \times G$.
\end{prob}

\begin{prob}
For $X$ a subgroup of $Y$, recall
that the index of $X$ in $Y$ is the number of equivalence classes of cosets
of $X$ in $Y$. Denote this by $(Y:X)$. If $L$ is a subgroup of $H$ which
is a subgroup of $G$, show that
$$(G:H)(H:L) = (G:L)$$
\end{prob}

\begin{prob}
Let $H$ be a subgroup of $G$. Show that the action of $G$ on the set of equivalence
classes of cosets $G/H$ defines a homomorphism $\phi:G \to S_{G/H}$. Show that the
kernel $K$ of $\phi$ is contained in $H$. If $H$ is {\em normal}, then 
show $K = H$. Using $\phi$ show that if the index of $H$ in $G$ is finite, then
the index of $K$ in $G$ is finite.
(Recall that $H$ is {\em normal} if for any $g \in G$, $gHg^{-1} = H$
as subsets of $G$)
\end{prob}

\begin{prob}
Show that
$$\Z/m\Z \times \Z/n\Z \cong \Z/mn\Z$$
if and only if $n$ and $m$ are coprime.
\end{prob}

\begin{prob}
Show that $S_2 \cong D_1 \cong \Z/2\Z$. Also, show that
$S_3 \cong D_3 \cong GL_2(\Z/2\Z)$.
\end{prob}

\begin{prob}
Suppose $A$ is an Abelian group (i.e. $gh = hg$ for all $g,h \in A$). Fix an integer
$n$ and let $\phi_n:A \to A$ be defined by $\phi_n(g) = g^n$. Show $\phi_n$ is
a homomorphism. Suppose $A$ is finite and the order of $A$ is coprime with $n$.
Show $\phi_n$ is an isomorphism.
\end{prob}

\begin{prob}
An {\em automorphism} of $G$ is an isomorphism from $G$ to itself. Show that
the collection of automorphisms of $G$ is itself a group, denoted $\text{Aut}(G)$.
\end{prob}

\begin{prob}[Hard]
If $g \in G$, the map $\phi_g:G \to G$ defined by $\phi_g(h) = ghg^{-1}$ defines
an automorphism of $G$ (check!). Such automorphisms are called {\em inner
automorphisms}. Show that the map $G \to \text{Aut}(G)$ defined by $g \to \phi_g$
is a homomorphism, and therefore that the set of inner automorphisms is a group.
Show in fact that it is a {\em normal} subgroup of $\text{Aut}(G)$. The
quotient group $\text{Aut}(G)/\text{Inn}(G)$ is called the group of {\em outer
automorphisms of $G$}. 
\end{prob}
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