define alternating groups in 4.5

circle.tex

@@ -2537,7 +2537,7 @@ \section{Interlude: combinatorics of permutations}
   However, in~\cref{cor:sign-defined} below, we'll show that 
   the \emph{parity} (odd/even) of the number of transitions is invariant.}
 
-\begin{xca}
+\begin{xca}\label{xca:factorial}
   Show that there are $n!$ permutations of a finite set of cardinality $n$, where $n! \defeq \fact(n)$ is the usual notation for the factorial function.
 
   \emph{Hint:} One way (not the only one) is to construct bijections

group.tex

@@ -477,8 +477,8 @@ \subsection{First examples}
 
   \item\label{ex:permgroup}
     If $n:\NN$, then the \emph{permutation group of $n$ letters}
-    (also known as the \emph{symmetric group of order $n$}) is%
-    \glossary(918Sigma2){$\protect\SG_n$}{symmetric group of order $n$,
+    (also known as the \emph{symmetric group of degree $n$}) is%
+    \glossary(918Sigma2){$\protect\SG_n$}{symmetric group of degree $n$,
       \cref{ex:groups}\ref{ex:permgroup}}\index{symmetric group}%
     \index{group!symmetric group}
     \[
@@ -1831,7 +1831,28 @@ \section{The sign homomorphism}
   subset with the standard $2$-element set,
   and using the pointedness of $\B\mu$.
 \end{definition}
-Something interesting happens when we consider 
+Not only does the notion of a sign ordering allow us to define the
+sign homomorphism, we also get a new family of examples of groups:\footnote{%
+  We'll study this construction more generally later in~\cref{subsec:ker}:
+  in these terms $\AG_n$ is the \emph{kernel} of the sign homomorphism.}
+\begin{definition}\label{def:alternating-groups}
+  For any $n:\NN$, we define the \emph{alternating group of degree $n$}
+  to be\index{alternating group}\index{group!alternating group}%
+  \glossary(An){$\protect\AG_n$}{alternating group of degree $n$,
+      \cref{def:alternating-groups}}
+  \[
+    \AG_n \defeq \mkgroup\Bigl(\sum_{A:\BSG_n}\Bsgn(A),
+      (\bn n, \Bsgn_\pt(\sh_{\SG_2}))\Bigr),
+  \]
+  \ie the shapes of $\AG_n$ are \emph{sign ordered $n$-element sets},
+  and the designated shape is $\bn n$ with the sign ordering coming
+  from the usual total ordering.
+
+  The symmetries in $\AG_n$ are called \emph{even permutations}.%
+  \index{permutation!even}
+\end{definition}
+
+Something interesting happens when we consider
 permutations on other shapes in $\BSG_n$,
 \ie arbitrary $n$-element sets $A$.
 The same map, $\Bsgn$, can be considered as a map $\BAut(A) \to \BSG_2$,
@@ -1852,7 +1873,9 @@ \section{The sign homomorphism}
   Otherwise, the \emph{sign} of $\sigma$ is $\pm1$ according to whether
   $\Bsgn_\div(\sigma)$ swaps the elements of the $2$-element 
   set $\Bsgn_\div(A)$, or not.
-  We write $\sgn(\sigma):\set{\pm1}$ for the sign of $\sigma$.
+  We write $\sgn(\sigma):\set{\pm1}$ for the sign of $\sigma$,
+  and call $\sigma$ \emph{even}\index{even} if $\sgn(\sigma)=1$,
+  and \emph{odd}\index{odd} otherwise.
 \end{definition}
 For permutations of the standard $n$-element set, this is the same as the value
 $\Usgn(\sigma) : \USG_2$. Note that $\sgn$ defines an abstract homomorphism from
@@ -1973,6 +1996,12 @@ \section{The sign homomorphism}
   (and thus the sign) is preserved, as seen in~\cref{fig:permutation-crossings-isotopy}.
 \end{remark}
 
+\begin{xca}
+  Recall from~\cref{xca:factorial} that there are $n!$
+  permutations in $\SG_n$.
+  Show that there are $n!/2$ even permutations for $n\ge 2$.
+\end{xca}
+
 \section{Infinity groups (\texorpdfstring{\inftygps}{∞-groups})}
 \label{sec:inftygps}