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\begin{document}
\begin{center}
    Math 390C (Geometry in Group Theory)
    \\
    Daniel Allcock
    \\
    Homework 2, due Friday Feb 7, 2020
\end{center}

Everyone should solve and turn in
problems marked ``everyone''.  Undergrads
should also solve and turn in problems marked
``undergrads'' {\bf PLUS ONE MORE OF THEIR CHOICE}.  
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}
    In class we discussed how the rotation group
    $A_5$ of the dodecahedron has preimage ``$2A_5$''
    of order~$120$
    in the $3$-sphere.  And we defined
    the 600-cell as the convex hull (in~$\R^4$) of those
    120 points.  For each $\theta\in[0,\pi]$,
    write $S_\theta$ for the ``spherical shell at angle~$\theta$
    from~$1$''.  Formally, this means the set of $g\in S^3$
    such that the angle between $1$ and $g$ (regarded as
    vectors in~$\R^4$) is~$\theta$.  Each shell is a $2$-sphere,
    except for $S_0$ and $S_\pi$ which degenerate to points.

    Find the numbers of points of $2A_5$ in each $S_\theta$.  
    Conjugation in $2A_5$ permutes the points in
    each shell.  Find the sizes of the orbits.
\end{problem}

\begin{problem}
    \everyone
    This problem demonstrates a sometimes-seemingly-magical
    way  to prove transitivity of a group action without
    having to get grubby and actually write down explicit
    transformations.

    Recall the $E_8$ lattice from hw1.  You showed there
    that its isometry group $W$ acts transitively on
    the norm~$2$ vectors (``roots'').  Prove that
    the root stabilizer acts transitively on the set of
    roots orthogonal to it.

    Do this as follows: Let $V$ (``visible'') 
    be the group generated
    by permutations
    of coordinates and the transformations that negate any
    even number of coordinates.  Check that $V\leq W$.
    Under $V$, there are two orbits on roots.  Choose a root,
    say $r$, and find the sizes of the orbits of
    the stabilizer $V_r$ on
    the set of roots orthogonal to~$r$.  Then do the same with 
    a root $s$ from the other $V$-orbit.  The
    $W$-stabilizer $W_t$ of any root~$t$ contains a conjugate of
    $V_r$ and a conjugate of $V_s$.  Use this 
    and the orbits sizes to show
    that $W_t$ is transitive on the roots orthogonal to~$t$.
\end{problem}

\begin{problem}
    This problem is mostly preparation for the next problem.
    The $D_n$ lattice means the subset of $\Z^n$ with even
    coordinate sum.  We equip it with the standard inner product.
    The very definition makes clear that $\Aut(D_n)$ contains
    the signed permutations of coordinates.  When
    $n\neq4$, prove
    that there are no more automorphisms.
\end{problem}

\begin{problem}
    \undergrad
    We work with  the $E_8$ lattice and continue using
    the notation above.  Prove $|W|=696,729,600$.  (Hint: this is
    an iterated orbit-stabilizer problem.)
    %From transitivity
    %on roots, and from the transitivity of the root 
    %stabilizer on the roots orthogonal to it, you know that
    %$$
    %|W|=240\cdot126\cdot|\hbox{pointwise stabilizer of 2 orthogonal roots}|
    %$$
    %Figure out the third term.)
\end{problem}

\begin{problem}
    \everyone
    This problem demonstrates another ``magical''
    way to prove transitivity without getting your hands grubby.
    We continue with $E_8$.  Prove that $W$ acts transitively
    on the set of norm~$4$ lattice vectors.  

    This is not hard to do directly by finding the $V$-orbits
    and seeing that the reflections in specific roots mix these
    orbits together.  But I want you to do this the magical way.
    First count the number of norm~$4$ vectors.  Then consider
    the specific example $v=(2,0,0,0,0,0,0,0)$.  Prove that
    the $W$-stabilizer of~$v$ is the same as its
    $V$-stabilizer, which has order $2^6 7!$.
    (Hint: consider the set of norm~$4$
    lattice vectors having the same image as~$v$ in $E_8/2E_8$.)
    Conclude that $v$'s orbit consists of all the norm~$4$
    vectors.
\end{problem}

\begin{problem}
    This problem develops a way to search for maximal subgroups
    of a finite group~$G$, more systematic than you probably
    used for~$A_5$ on hw1.  

    Prove: every finite group~$M$ has a normal subgroup which
    is a direct product of some number of copies of a
    single simple group.  

    Taking $M$ to be a maximal
    subgroup of~$G$, conclude that every maximal subgroup
    $G$ can be described as the normalizer of a subgroup
    of this special form.  

    (For $A_5$ this gives that every maximal subgroup
    arises as the normalizer of a $2$, or a $2\times2$,
    or a~$3$ or a~$5$.  All of these normalizers
    are indeed maximal
    except for the first.)
\end{problem}

\begin{problem}[Action of $\PSL_2(\F)$ on $\P^1\F$]
    \everyone
    Let $\F$ be a field.  Obviously $\PSL_2(\F)$ acts
    on~$\P^1\F$.  Prove that it acts $2$-transitively
    (that is, transitively on ordered pairs of distinct
    points).  Prove that it might or might not act 
    $3$-transitively, depending on~$\F$.  Find a 
    necessary and sufficient condition on~$\F$ for it to
    act $3$-transitively.
\end{problem}

\begin{problem}[Tutte's $8$-cage]
    ``Tutte's $8$-cage'', to give it its 19th-century name,
    is a graph~$\Gamma$ you can download from the course webpage.
    Prove it is the incidence graph of the ``duads'' and
    ``synthemes'' from a set~$\Omega$ of size~$6$.  Here
    a duad means a $2$-point set, and a syntheme means a
    partition of $\Omega$ into three duads.  A duad is
    incident to a syntheme just if it is one of those three
    duads.  (One can also think of duads and synthemes
    as elements of~$S_6$, with cycle shapes $21111$ and $222$.)

    Let $G$ be the automorphism group of~$\Gamma$  (ignoring
    vertex color).  Obviously it contains~$S_6$.  
    Prove that~$G$ is larger, of order $1440$. (Hint: iterated
    orbit-stabilizer.)
    Prove that $A_6$ is normal in~$G$, so there is a natural
    map $G\to\Aut(A_6)$.
    Prove that the $G$-centralizer of~$A_6$ is trivial, so this map is
    injective (hint: anything that
    commutes with
    a subgroup $T\iso2\times2$ of~$A_6$  must preserve
    the fixed-point set of
    $T$ in~$\Gamma$).
    Find a way to express $\Gamma$ entirely in terms of
    the subgroups $2\times2$ of $A_6$, so that $\Aut(A_6)$
    embeds in~$G$.  Conclude $G=\Aut(A_6)$.
\end{problem}

%\begin{problem}[``The icosahedron exists only over certain fields.'']
%    Suppose $\F$ is a finite field other than $\F_2$.
%    Prove that $A_5$ embeds in $\PSL_2(\F)$ if and only
%    if $\F$ contains a square root of~$5$.
%    (Hint: ``build'' an icosahedron in 
%\end{problem}
%
%\begin{problem}
%    By the previous problem, $A_5$ embeds in $\PSL_2(\F_{11})$.
%\end{problem}


%\begin{problem}[``Finite hyperbolic planes]
%    Let $\F$ be a finite field of odd order, and let $V$
%    be a $3$-dimensional vector space over~$\F$, equipped with
%    a nondegenerate symmetric bilinear form.  Define $\O(V)$
%    as the group of 
%\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}
