starting working 5.3 over

actions.tex

@@ -652,18 +652,26 @@ \section{Subgroups}
 \subsection{Subgroups through $G$-sets}
 
 The idea is that a $G$-set $X$ picks out those symmetries in $G$
-that keep the point of $X(\sh_G)$ in place. For this to work well
+that keep a chosen point of $X(\sh_G)$ in place. For this to work well
 we need to point $X(\sh_G)$ and $X$ must be transitive.
 %so that the set of symmetries that are picked out is closed under composition and reverse.
 
 \begin{definition}\label{def:set-of-subgroups}
-  For any group $G$, define the type of \emph{subgroups of $G$} as
-  \index{type!of subgroups of a group}
-%  \glossary(SubG){$\protect{\Sub_G}$}{type of subgroups of $G$} ERROR???
-  $$\typesubgroup_G\defequi\sum_{X:\BG\to\Set}{\,}X(\sh_G)
-  \times\istrans(X).$$
-  The \emph{underlying group} of the subgroup $(X,x,!) : \Sub_G$ is
-  $$\mkgroup \bigl(\sum_{z:BG}X(z),(\sh_G,x)\bigr).\qedhere$$
+  For any group $G$, define the type of \emph{subgroups of $G$} as%
+  \index{type!of subgroups of a group}%
+  \glossary(SubG){$\protect\Sub(G)$}{type of subgroups of $G$}
+  \[
+    \Sub(G)\defequi\sum_{X:\BG\to\Set}{\,}X(\sh_G)
+    \times\istrans(X).
+  \]
+  The \emph{underlying group} of the subgroup $(X,x) : \Sub(G)$ is\footnote{%
+    To lighten the notation, we leave out the proof that $X$ is transitive.
+    (Otherwise, we would write $(X,x,!):\Sub(G)$.)
+    In~\cref{rem:notationsubgroup} below we'll set out further notational conveniences
+    regarding subgroups.}
+  \[
+    \mkgroup \biggl(\sum_{z:BG}X(z),(\sh_G,x)\biggr).\qedhere
+  \]
 \end{definition}
 
 \begin{xca}\label{xca:group-Xx!}
@@ -687,18 +695,18 @@ \subsection{Subgroups through $G$-sets}
 $R(\Sloop) \defis \etop\zs$, see \cref{def:RtoS1}.
 Again we point by $0: R(\base)$ and transitivity of $R$ is obvious.
 The only symmetry that keeps $0$ in place is $\refl{\base}$,
-since $R(\Sloop)= \zs$ if and only if $k=0$.
+since $R(\Sloop^k)(0) = \zs^k(0) = k = 0$ if and only if $k=0$.
 Again, no surprise in view of the results in \cref{sec:symcirc}
 identifying $R$ as the universal \covering over $\Sc$.
 
-The following result is analogous to the fact that $\Sub_T$ is
+The following result is analogous to the fact that $\Sub(T)$ is
 a set for any type $T$, see \cref{xca:subtypes-set}. It captures
 that the essence of picking out symmetries (or picking out elements
 of a type), is a predicate, like $R_m(p)(0)=0$ above.
 
 \begin{lemma}
   \label{lem:SubGisset}%
-  The type $\typesubgroup_G$ is a set, for any group $G$.   
+  The type $\Sub(G)$ is a set, for any group $G$.
 \end{lemma}
 \begin{proof}
 Let $G$ be a group, and let $X$ and $X'$ be transitive $G$-sets with