minor in subgroups

actions.tex

@@ -451,8 +451,8 @@ \subsection{Transitive $G$-sets}
 \begin{marginfigure}
   \noindent\begin{tikzpicture}
     \pgfmathsetmacro{\len}{1}
-    \node[vertex,label=above:$x$] (n1) at (0:\len) {};
-    \node[vertex] (n2) at (120:\len) {};
+    \node[vertex] (n1) at (0:\len) {};
+    \node[vertex,label=above:$x$] (n2) at (120:\len) {};
     \node[vertex] (n3) at (240:\len) {};
     \begin{scope}[every to/.style={bend right=22}]
       % generator a
@@ -682,12 +682,12 @@ \subsection{Subgroups through $G$-sets}
 As an example, recall from \cref{def:RmtoS1} the $\Sc$-set
 $R_m : \Sc\to\Set$ defined by $R_m(\base) \defeq \bn m$ and
 $R_m(\Sloop) \defis \etop\zs$. Here $m>0$ so that we can point
-$R_m$ by $0: R_m(\base)$.\footnote{Any element of $\bn m$ would do.} 
+$R_m$ by $0: R_m(\base)$.\footnote{Any element of $\bn m$ would do.}
 Transitivity of $R_m$ is obvious.
 Which symmetries $p: \base\eqto\base$ are picked out by $R_m$?
 Those that keep the point $0: R_m(\base)$ in place, that is,
 those that satisfy $R_m(p)(0)=0$, \ie $p=\Sloop^{mk}$ for some integer $k$.
-Given $\alpha_m$ in \cref{con:psi-alpha-m}, it should not come as a 
+Given $\alpha_m$ in \cref{con:psi-alpha-m}, it should not come as a
 surprise that these are precisely the symmetries picked out by $\dg{m}$.
 
 The case of $m=0$ connects to another old friend, the $\Sc$-set
@@ -720,27 +720,29 @@ \subsection{Subgroups through $G$-sets}
 
 \begin{example}
   \label{exa:fix1subSGn}%
-Consider the group $\SG_n$ (\cref{ex:groups}\ref{ex:permgroup}) 
-for given $n>0$. For any $k:\bn n$, define
-the $\SG_n$-set $X_k : \BSG_n  \to\Set$ by $X_k(A,!)\defeq A$ for any
-$A:\FinSet_n$. Then $X_k$ is obviously transitive. We point $X_k$ by
-$k: X_k(\sh_{\SG_n}) \jdeq \bn n$.\footnote{Here the choice of the point
+Consider the symmetric group $\SG_n$ from~\cref{ex:groups}\ref{ex:permgroup},
+for some $n>0$. It has a canonical action,
+the $\SG_n$-set $X : \BSG_n \to\Set$ given by $X(A,!)\defeq A$
+for any $A:\FinSet_n$, which is obviously transitive.
+For any $k:\bn n$, we can point $X$ by
+$k: X(\sh_{\SG_n}) \jdeq \bn n$.\footnote{The choice of the point
 does matter for the symmetries that are picked out.}
-Thus we have $(X_k,k,!):\Sub_{\SG_n}$.
+Thus we have $(X,k):\Sub(\SG_n)$.
 The symmetries that are picked out are those $\pi : \bn n \eqto \bn n$
-that satisfy $(\pi \cdot_{X_k}k) = k$.\footnote{%
-This uses the alternative notation for the group action of $X_k$
+that satisfy $(\pi \cdot_X k) = k$.\footnote{%
+This uses the alternative notation for the group action of $X$
 introduced in \cref{def:Gset}.}
 In other words, $\pi$ keeps $k$ in place and can be any permutation
-of the other elements of $\bn n$. From the next \cref{xca:n-is-ptd-n+1}
-we get that the underlying group of each $(X_k,k,!)$ 
+of the other elements of $\bn n$.
+From the next~\cref{xca:n-is-ptd-n+1}
+we get that the underlying group of each $(X,k)$
 is isomorphic to $\SG_{n-1}$.
 \end{example}
 
-\begin{xca} \label{xca:n-is-ptd-n+1}
-Give an equivalence from the type of
-$n$-element sets to the type of pointed $(n{+}1)$-element sets.
-Hint: use \cref{xca:finsets-decidable}.
+\begin{xca}\label{xca:n-is-ptd-n+1}
+  Give an equivalence from the type of
+  $n$-element sets to the type of pointed $(n{+}1)$-element sets.
+  Hint: use~\cref{xca:finsets-decidable}.
 \end{xca}
 
 \begin{xca} \label{xca:A-is-A-1+1}
@@ -751,6 +753,7 @@ \subsection{Subgroups through $G$-sets}
 For yet another example, consider the cyclic group $\CG_6$ of order $6$; perhaps visualized as the rotational symmetries of a regular hexagon,  \ie the rotations by $2\pi\cdot m /6$, where $m=0,1,2,3,4,5$.
 The symmetries of the regular triangle (rotations by $2\pi\cdot m/3$, where $m=0,1,2$) can also be viewed as symmetries of the hexagon.
 Thus there is a subgroup of $\CG_6$ which, as a group, is isomorphic to $\CG_3$.\marginnote{Make a TikZ drawing of the hexagon and triangle inscribe in it.}
+%LINK TO CH 3 AND SQUARE ROOT OF 6 BUNDLE
 
 \begin{example}
   \label{exa:C3subC6}%

macros.tex

@@ -859,9 +859,9 @@
 %    sep=0, outer sep=0, circle]%
 %    {$\bullet$};}}%
 \newcommand*{\base}{{\sbt}}%point in circle
-\newcommand*{\Cloop}{\circlearrowleft}% loop in circle \InfCycSet
-\newcommand*{\Sloop}{\circlearrowleft}% loop in circle \Sc
-\newcommand*{\qedge}{\curvearrowright}% edge in graph quotient
+\newcommand*{\Cloop}{\mathop\circlearrowleft}% loop in circle \InfCycSet
+\newcommand*{\Sloop}{\mathop\circlearrowleft}% loop in circle \Sc
+\newcommand*{\qedge}{\mathop\curvearrowright}% edge in graph quotient
 
 \newcommand*{\conncomp}[2]{{{#1}_{\left(#2\right)}}}%
 \newcommand*{\univcover}[2]{{{#1}^0_{\left(#2\right)}}}%