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\begin{center}
MATH 776 HOMEWORK 12 \\
{\it Semester II, 2006-2007} \\
{\it Due April 26}
\end{center}


\bigskip

\bigskip
\textbf{57.} Assume that $\Sigma$ in $\LL$ is complete and
has a finite model $\AAAA$.  Then all models of $\Sigma$ are
isomorphic to $\AAAA$.  $|\LL|$ can be arbitrary here.
\emph{Hint.} Say $\BBBB\models \Sigma$.  First note that 
$|A| = |B|$.  List $A$ as $\{a_i : i < n\}$ and then inductively
chooose $b_0, b_1, \ldots \in B$ so that 
$(\AAAA, a_i)_{i < j} \equiv (\BBBB, b_i)_{i < j} $.
\emph{Remark.} This is the finite version of the uniqueness of
saturated models.


\bigskip
\textbf{58.} Let $\GGGG$  be a group.  For $a \in G$, let
$o(a)$ be the order of $a$ (a finite number or $\infty$).
$\GGGG$ is a \emph{torsion group} iff $o(a) < \infty$ for all $a \in G$,
and $G$ has \emph{finite exponent} iff for some $n \in \omega$,
$o(a) \le n$ for all $a \in G$.
Let $\UU$ be a countably incomplete ultrafilter on $I$.
Prove that the following are equivalent:
\begin{itemizz}
\item[1.] $\GGGG^I /\UU$ is a torsion group.
\item[2.] $\GGGG^I /\UU$ has finite exponent.
\item[3.] $\GGGG$ has finite exponent.
\end{itemizz}
\emph{Hint 1.} You can use 
the fact that the ultrapower is an
$\aleph_0$--saturated elementary extension;
then you don't need to know what an ultrapower is.
\emph{Hint 2.} You can argue directly from the definition of the
ultrapower; then you don't need to know any logic.

\bigskip
\textbf{59.} A group $G$ is \textit{simple} iff $G$ has no normal subgroups
except for $G$ and $\{1\}$.
Prove
\begin{itemizz}
\item [a.]
Every ultraproduct of non-simple groups is non-simple.
\textit{Hint.} Add a predicate for a normal subgroup.
\item [b.] 
$\prod_{n\in\omega} A_n / \UU$ is not simple if $\UU$ is non-principal.
\textit{Hint.} You need to multiply $k$ conjugates of $(1,2)(3,4)$
to get the permutation
$(1,2)(3,4)\cdots (4k \mathord{-} 1, 4k)$,
\item [c.] 
For some finite simple groups $G_n$, 
$\prod_{n\in\omega} G_n / \UU$ is simple and infinite for all
non-principal $\UU$.
\textit{Hint.} Make all $G_n$ simple for some ``uniform reason'';
for example $\mathit{PSL}(2, K)$ is simple for all fields $K$ with 
$4$ or more elements.  $\mathit{PSL}(2, K) = S/Z(S)$,
where $S = \mathit{SL}(2, K)$ is all $2\times 2$ matrices over $K$
with determinant $1$.
\end{itemizz}
\emph{Remark.} Since you're talking about ultraproducts, it's safe
to be a bit sloppy about exactly what $\LL$ is.


\bigskip
\textbf{60.} Let
$\LL = \{<\}\cup \{c_i: i < \omega\} \cup \{e_i: i < \omega\} $ and
let $\Sigma$ be the theory of dense total orders without endpoints
(expressed just using $<$), plus
\[
c_0 < c_1 < c_2 < \cdots < e_2 < e_1 < e_0 \ \ .
\]
Describe the countable saturated model,
the countable atomic model (only the isolated 
types are realized),
and all the other countable models.  How many are there?
 
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