spell out pointwise operations

Author: UlrikBuchholtz

group.tex

@@ -903,14 +903,19 @@ \section{Abstract groups}
 provides a different source of examples.
 
 \begin{xca}\label{xca:abstract-group-of-maps}
-Let $\agp G$ be an abstract group with underlying set $S$.
-Let $X$ be a set. Show that the set $X\to S$ of functions
-from $X$ to $S$, together with pointwise operations induced by
-$\agp G$, forms and abstract group which is abelian if and only
-if $\agp G$ is.
+  Let $\agp G$ be an abstract group with underlying set $S$
+  and neutral element $e$.
+  Let $X$ be a set. Show that the set $X\to S$ of functions
+  from $X$ to $S$, together with the ``pointwise operations''
+  inherited from $\agp G$,
+  \begin{alignat*}2
+    \cst{e} &\jdeq (x \mapsto e) &\qquad &\text{(neutral element)}\\
+    f \cdot g &\defeq (x \mapsto f(x) \cdot g(x)) &\qquad &\text{(multiplication)},
+  \end{alignat*}
+  forms an abstract group, which furthermore is abelian if and only
+  if $\agp G$ is.
 \end{xca}
 
-
 We leave the study of abstract groups for now;
 in~\cref{ch:absgroup} we'll
 show that the $G \mapsto \abstr(G)$ construction