definition group action

inputs/lecture_15.tex

@@ -1,5 +1,31 @@
 \lecture{15}{2023-12-05}{}
 
+Recall:
+\begin{definition}+
+    Let $X$ be a set.
+    A \vocab{group action} of a group $G$ on $X$
+    is a function
+    $\alpha\colon G \times X \to X$
+    such that
+    \begin{itemize}
+        \item $\forall x \in X.~\alpha(1_G,x) = x$,
+        \item $\forall g,h \in G, x \in X.~\alpha(gh,x) = \alpha(g,\alpha(h,x))$.
+    \end{itemize}
+
+    Often we will abbreviate $\alpha(g,x)$ as $g\cdot x$.
+\end{definition}
+\begin{remark}+
+    Group actions of a group $G$ on a set $X$
+    correspond to group-homomorphisms
+    $G \to \Sym(X)$.
+    Indeed for a group action $\alpha\colon G \times X \to X$
+    consider
+    \begin{IEEEeqnarray*}{rCl}
+        G&\longrightarrow & \Sym(X) \\
+        g&\longmapsto & (x \mapsto g \cdot x).
+    \end{IEEEeqnarray*}
+\end{remark}
+
 \begin{theorem}[The Boundedness Theorem]
     \yalabel{Boundedness Theorem}{Boundedness}{thm:boundedness}