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\begin{document}
\begin{center}
    Math 390C (Geometry in Group Theory)
    \\
    Daniel Allcock
    \\
    Homework 1, due Friday Jan 31, 2020
\end{center}

Everyone should solve and turn in
problems marked ``everyone''.  Undergrads
should also solve and turn in problems marked
``undergrads''.  The remaining problems
are optional; everyone should try 
them, and solve them if
they seem rewarding.  And grad
students should try the ``undergrad'' problems
in a similar way.  
Students not enrolled should also try the problems
and write up or talk to me about any they find
interesting or troubling.

\begin{problem}
\everyone
    Classify the maximal subgroups $M$ of $A_5$
    up to conjugacy.  (Hint: use Sylow's theorem
    in~$M$.)  Describe the subgroups as stabilizers
    of objects expressed in terms of a $5$-point
    set~$\Omega$.  For example, ``the stabilizer of
    an unordered pair of distinct points in~$\Omega$''.
    These objects are what I called the
    elements of the geometry (of~$A_5$).
\end{problem}

If $F$ is a field, then projective $n$-space $\P^n\!F$
means
the set of lines through the origin in the vector
space $F^{n+1}$.  $\P^1\!F$ is called the projective
line and $\P^2\!F$ is called the projective plane.

\begin{problem}[Fractional linear transformations]
\everyone
    Let $F$ be a field.  Then the elements of
    $\P^1\!F$ are sometimes written as follows.
    The line spanned by $(n,d)\neq(0,0)$ is indicated
    by $n/d\in F$.  The line spanned by $(n,0)$ is
    indicated by the special symbol~$\infty$.

    Prove that $\GL_2(F)$ acts on $\P^1\!F$ by what
    are called \defn{linear fractional transformations},
    namely
    $$
    z\mapsto\frac{az+b}{cz+d}
    $$
    for $a,b,c,d\in F$ with $ad-bc\neq0$.
    Here the variable $z$ varies over
    $F\cup\{\infty\}$, and the formula is
    interpreted as $\infty$ or~$0$ if the denominator
    is~$0$ or~$\infty$.
\end{problem}

\begin{problem}
\undergrad
    Build a physical model of a dodecahedron with
    the faces labeled $0,1,2,3,4,5,\infty$ (opposite
    faces given the same label), such that orientation-preserving
    isometries
    of the dodecahedron act by fractional linear
    transformations of~$\P^1\F_5$.

    [You know ahead of time that such a model must
    exist, because (1) $\PSL_2\F_5$ is nonsolvable
    with order~$60$,
    (2) there is only one such group, namely~$A_5$,
    and (3) $A_5$
    has only one way to act transitively 
    on~$6$ points.]
\end{problem}

    Recall that a group $G$ acting transitively on
    a set~$X$ acts \defn{primitively} if there is
    no $G$-invariant partition of~$X$ other than the
    obvious ones (the partition into singletons
    and the ``partition'' into a single set).
    We saw in class that this is another way to say
    that the point stabilizer is a maximal subgroup.
    In this case, $X$ is one family of elements of the
    geometry (of~$G$).
    
\begin{problem}
    \optional
    Prove that the isometry group of Euclidean $n$-space
    acts primitively on the points of~$\R^n$.  (Adapt the
    argument I hand-waved in class for the hyperbolic plane.
    Also, the
    problem as stated is wrong; fix it.)
\end{problem}

\begin{problem}
    \undergrad
    The projective plane over~$\F_2$ has $7$ points and
    $7$ lines, corresponding to the nonzero vectors in
    $\F_2^3$ and its dual space.  $\GL_3(2)$ acts 
    transitively on the points, hence has order divisble
    by~$7$, hence contains a cyclic group of order~$7$.
    Therefore it is possible to draw a picture in the plane
    where the points are represented by $7$ white dots and
    the lines by $7$ black dots, dots are joined just if the
    corresponding point and line are incident, and the whole
    picture is invariant under an order~$7$ rotation.
    Do it.
\end{problem}

\begin{problem}[Icosahedron exists, ``soft'' argument]
\optional
    Begin with a regular octahedron, and label the edges
    with arrows, such that each face has 3 arrows
    going around it cyclically.  Imagine 12 ants, one
    starting at the tail of each edge, walking along
    the edges  at  speed~$1$ until
    reaching the end.  Taking their convex hull is gives
    a $1$-parameter family of polyhedra.  Use the Intermediate
    Value Theorem to show that one member of this
    family 
    has 20 
    equilateral triangles for its faces.
    Use this and the fact that all vertices are the same
    distance from the origin to show that any face
    can be sent to any other by an isometry.
    (Argument from Coxeter's \emph{Regular Polytopes}.)
\end{problem}

%\begin{problem}[Icosahedron exists, ``Fiat Lux'' style]
%    Let $I$ be the convex hull of the twelve points
%    got from $(0,1,\phi)$ by permuting and changing
%    signs of coordinates.  Here $\phi$ is the golden
%    ration $(1+\sqrt5)/2$.  Prove that 
%\end{problem}

A \defn{lattice} $L$ means a free $\Z$-module equipped with
a symmetric biliner form $L\times L\to\Q$.  Often we define
$L$ as a subset of $\R^n$, in which case we mean the standard
inner product unless otherwise stated.  Usually
we call the pairing the dot product and write $\cdot$.
The \defn{norm} of a vector~$x$ means $x\cdot x$ 
(NOT $\sqrt{x\cdot x}$).
We call $L$ \defn{integral} if all inner products are
integers.  An \defn{isometry} of a lattice $L$ (or
vector space equipped with a symmetric bilinear form)
means an invertible self-map that preserves inner products.
$\Aut(L)$ means the group of all isometries of~$L$.

\begin{problem}
    \everyone
    Suppose $L$ is an integral lattice and $r\in L$.
    The \defn{reflection} $R_r$ in~$r$
    means the linear map on the enclosing vector space
    $$R_r(x)= x-2\,\frac{x\cdot r}{r^2}r$$
    Check that this
    preserves inner products,
    negates $r$ and fixes $r^\perp$ pointwise.
    If  $r^2\in\{\pm1,\pm2\}$ 
    then $R_r\in\Aut(L)$.  This gives a way to build
    lots of isometries without having
    to write down matrices; for example
    the next couple of problems.
\end{problem}

    We write \defn{$\gend{\ldots}$} for the group
    generated by $\ldots$

\begin{problem}
    \optional
    Suppose $L$ is an lattice and $r,s\in L$ have
    inner product~$\pm1$.  Then they are equivalent
    under $\gend{R_r,R_s}$.
\end{problem}

\begin{problem}
    The $E_8$ lattice is one of the most remarkable
    objects in mathematics.  It means
    $$
    \Bigl\{
        (x_1,\dots,x_8)\in({\textstyle\frac12}\Z)^8
        \Bigm|
        \hbox{$x_i-x_j\in\Z$ for all $i,j$, and $\sum x_i\in2\Z$}
        \Bigr\}
    $$
    Prove it is integral, find the norm~$2$ vectors,
    and prove that they
    are equivalent under lattice isometries.
\end{problem}

%\begin{problem}
%    \all
%    The $D_4$ lattice means the set of vectors in~$\Z^4$
%    with even coordinate sum.  Its dual lattice $D_4^*$
%    means the set of vectors in $\R^4$ having integer
%    inner products with all the lattice vectors.  
%
%    Each of $D_4$ and $D_4^*$ has $24$ minimal elements,
%    and each set of $24$ consists of the vertices of
%    a $24$-cell (the $4$-dimensional regular polytope).
%    The two $24$-cells arising this way are each
%    others' dual polytopes.
%\end{problem}
%
%\begin{problem}
%    \optional
%    Find the orbits of vectors of norm${}\leq12$
%    in the $D_4$ lattice, under its
%    isometry group.  (The subgroup of $\O(4)$
%    that preserves the $D_4$
%    lattice.  This is slightly larger than the visible
%    $2^3:S_4$.)
%\end{problem}
%
%\begin{problem}
%    \optional
%    The $E_6$ lattice means the $6$-dimensional
%    lattice generated by 
%    the vectors
%    $$
%    \begin{matrix}
%        &&\hss\frac12({+}{+}{+}{+}{-}{-}{-}{-})\hss
%        \\
%        0{-}{+}00000
%        &
%        00{-}{+}0000
%        &
%        000{-}{+}000
%        &
%        0000{-}{+}00
%        &
%        00000{-}{+}0
%    \end{matrix}
%    $$
%    where $+$ and $-$ are abbreviations for $+1$
%    and $-1$. Eight coordinates turn out to be
%    better than~$6$, because $S_6$ symmetry is
%    visible.  (There is additional symmetry too.)
%    Classify the orbits of ordered
%    tuples of mutually orthogonal minimal vectors,
%    up to isometries.
%
%    [This is long because you have to figure out
%    a lot of things along the way.  Doing this 
%    exact problem in grad skool taught me a lot.
%    We will see more about $E_6$ later.]
%\end{problem}

\end{document}
