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\begin{document}

\noindent
{\large Representations of Groups.}

\smallskip

Groups were invented to understand the structure-preserving
permutations of a mathematical object. Consequently
most applications of group theory apply to situations where
the group elements are symmetries of some object
(or, structure-preserving 
permutations of the base set of some object.)
The study of groups from this viewpoint requires
{\it representing} the group elements as permutations,
and the subject is called {\it representation theory}.
A {\it permutation representation} of a group is a
way of representing the group as a group of permutations
of a set, while a {\it linear representation} of a group
is a way of representing the group as a group of invertible 
linear transformations of a vector space.

\begin{df} (G-set, representation, action)
Let $G$ be a group. A {\it $G$-set} is an algebra
$\lb X;G\rb = \lb X; \{\hat{g}(x)\;|\;g\in G\}\rb$ that has
one unary operation $\widehat{g}$ on $X$ for each $g\in G$, and
which satisfies the equational laws
\begin{enumerate}
\item[(i)] $\hat{1}(x) = x$ for all $x\in X$; and 
\item[(ii)] $\hat{g}(\hat{h}(x)) = \widehat{gh}(x)$.
\end{enumerate}

A {\it representation (or action) of $G$ on $X$} is a homomorphism 
$\varphi:G\rightarrow S_X$. The {\it kernel} of the representation
is ${\rm Ker}(\varphi)$.
\end{df}

\noindent
{\bf Exercise:} Show that if $\lb X;G\rb$ is a $G$-set, then 
the function $\varphi:G\rightarrow S_X:g\mapsto \hat{g}$
is a representation of $G$. Conversely, if 
$\varphi:G\rightarrow S_X$ is a representation of $G$ on $X$,
then show that by taking $\hat{g} = \varphi(g)$ we get 
a $G$-set $\lb X; \{\hat{g}(x)\;|\;g\in G\}\rb$.

\medskip

Based on this exercise, we say that two representations 
$\varphi:G\rightarrow S_X$ and $\phi:G\rightarrow S_Y$
are {\it isomorphic} if the corresponding $G$-sets
$\lb X;G\rb$ and $\lb Y;G\rb$ are isomorphic as 
algebras. This means exactly that there is a bijection 
$\alpha:X\rightarrow Y$ such that 
$\alpha\circ\varphi\circ\alpha^{-1} = \phi$.

\begin{df} (transitive, fixed point, regular, faithful)
%Assume that $\lb X;G\rb$ is a $G$-set.
The $G$-set $\lb X;G\rb$ 
is {\it transitive} is whenever $x,y\in X$
there is a function $\hat{g}$ such that $\hat{g}(x) = y$.
A {\it fixed point} of a function $\hat{g}$,
$g\in G$, 
is an element $x\in X$ such that $\hat{g}(x) = x$.
The $G$-set $\lb X;G\rb$ is {\it regular} if it is transitive and 
the only $\hat{g}$ that has a fixed point is $\hat{g} = \hat{1}$.
The $G$-set $\lb X;G\rb$ is {\it faithful} if the only function $\hat{g}$
that has every $x\in X$ as a fixed point is 
$\hat{g}=\hat{1} = {\sl id}_X$.
\end{df}

\vfill
\pagebreak

\noindent
{\bf Examples}

\begin{enumerate}
\item[(a)] $S_n$ acts on $\{1,2,\ldots,n\}$.
\item[(b)] $GL(n,{\BR})$ acts on ${\BR}^n$.
\item[(c)] $G$ acts on $G$ by left multiplication 
(the regular representation).
\item[(d)] $G$ acts on the left cosets $G/H$ of a subgroup
$H$ by left multiplication. (This means $\hat{g}(aH) = gaH$.)
\item[(e)] $G$ acts on $G$ by conjugation.
(This means $\hat{g}(h) = \gamma_g(h) = ghg^{-1}$.)
\item[(f)] ${\rm Aut}(G)$ acts on $G$.
\item[(g)] Dihedral groups act on regular polygons.
\item[(h)] $A_4$ acts on the regular tetrahedron by rotation.
(For this reason $A_4$ is called the {\it tetrahedral group}.)
\item[(i)] $S_4$ acts on the regular octahedron or cube by rotation.
(For this reason $S_4$ is called the {\it octahedral group}.)
\item[(j)] $A_5$ acts on the regular icosahedron or dodecahedron by rotation.
(For this reason $A_5$ is called the {\it icosahedral group}.)
\end{enumerate}

\smallskip

\noindent
{\bf Some easy questions}

\begin{enumerate}
\item[(a)] Is the usual representation of $S_n$ on 
$\{1,2,\ldots,n\}$ transitive? regular? faithful?
\item[(b)] Show that the regular representation is regular.
\item[(c)] Show that the regular representation is faithful.
\item[(d)] Show that kernel of the action of
$G$ on $G/H$ by left multiplication
is $\bigcap_{g\in G} gHg^{-1}$.
\item[(e)] Show that a representation is faithful
iff its kernel is $\{1\}$.
\item[(f)] Is the usual representation of $S_n$ isomorphic
to the regular representation of $S_n$?
\item[(g)] Is the action of ${\rm Aut}(G)$ on $G$
transitive? Is it faithful?
\end{enumerate}

\medskip

\noindent
{\bf Some medium-easy questions}

\begin{enumerate}
\item[(a)] Show that $S_n$ has no faithful 
representation on a set of $<n$ elements.
Show that the same is true for $A_n$.
\item[(b)] Is the representation of $D_n$ on the 
vertices of the $n$-gon isomorphic to the
representation of $D_n$ on the edges of the $n$-gon?
\item[(c)] Find a group with three nonisomorphic
faithful representations. (Hint: The smallest example has 12 elements.)
\end{enumerate}

\smallskip

\noindent
{\bf Some medium-hard questions}

\begin{enumerate}
\item[(a)] Show that the automorphism group 
of the $G$-set $\lb G/H; G\rb$ is isomorphic
to $N_G(H)/C_G(H)$. (See first section of Chapter 3.)
\item[(b)] Show that if $n\neq 6$, then any faithful 
representation of $S_n$ on $\{1,2,\ldots,n\}$ 
is isomorphic to the usual one.
\item[(c)] Find a faithful 
representation of $S_6$ on $\{1,2,\ldots,6\}$ that
is not isomorphic to the usual representation.
\end{enumerate}

\vfill
\pagebreak

\noindent
{\bf Some Basic Results}

\smallskip

\begin{df}
If $\lb X;G\rb$ is a $G$-set and $x\in X$, then the {\it orbit of $x$}
is the set $\orb(x)=\{y\;|\;\exists \hat{g}(y=\hat{g}(x))\}$.
The {\it stabilizer of $x$} is the set 
$G_x = \{g\in G\;|\;\hat{g}(x) = x\}$.
\end{df}

Note that, by definition, an orbit is nothing more than
a 1-generated (or {\it cyclic})
subalgebra of the $G$-set $\lb X;G\rb$.

\begin{th}
The relation $\equiv$ defined by ``$x\equiv y$ if $y$ is in the orbit
of $x$'' is an equivalence relation. The equivalence classes
are the orbits.

The subalgebras of the $G$-set $\lb X;G\rb$ are precisely 
the subsets of $X$ that are unions of orbits.

A subalgebra of $\lb X;G\rb$ is transitive if and only if
it is a single orbit.
\end{th}

\begin{th}
The stabilizer $G_x$ of any element $x\in X$ is a subgroup
of $G$. Elements $x, y\in X$ lie in the same orbit if and only
if their stabilizers are conjugate: 
\[\orb(x) = \orb(y) \Longleftrightarrow \exists g(G_y = gG_xg^{-1}).\]
The size of an orbit $\orb(x)$ is $[G:G_x]$.
\end{th}

\begin{cor}
If a finite group 
$G$ acts on a set $X$, then the number of elements in any orbit
divides $|G|$.
\end{cor}

\begin{th} (Structure Theorem for $G$-sets)
Any $G$-set is uniquely decomposable into 
a disjoint union of transitive $G$-sets (orbits), and any 
disjoint union of transitive $G$-sets is again a $G$-set.

Any transitive $G$-set $\orb(x)$ is isomorphic to the representation
of $G$ on the left cosets of $G_x$: $\lb G/G_x;G\rb$.
\end{th}

\begin{df}
Let $G$ act on itself by conjugation: $\hat{g}(x) = \gamma_g(x) = gxg^{-1}$.
An orbit $\orb(h)$ of an element $h\in G$ is called {\it the conjugacy
class} of $h$, and it is written $h^G$. 
The stabilizer $G_h$ of an element $h$ is
called the {\it centralizer of $h$ in $G$}, and it is written $C_G(h)$.
\end{df}

\noindent
{\bf Exercises:} 
\begin{enumerate}
\item[(a)]
Show that $|h^G| = [G:C_G(h)]$.
\item[(b)] Show 
that a subgroup of $G$ is normal if and only
if it is a union of conjugacy classes. 
\item[(c)] Find the sizes of the conjugacy classes
of elements in $A_5$. (Hint: Read Lemma 3.8 from the book.)
\item[(d)] Use the answer in (c) to prove that $A_5$ is
simple. (Any $N\lhd A_5$ must have order $|N| = n$ that
divides $|A_5| = 60$, but $n$ must also be a sum of sizes
of conjugacy classes. Show that the only values of
$n$ that work are $n=1$ and $n=60$.)
\end{enumerate}
\end{document}

