add 2 more problems to groups chapter

chapter-1-groups.tex

@@ -375,6 +375,24 @@ \subsection{Problems}
     % Note: Taken from 2012 Putnam A2
 \end{prob}
 
+\begin{prob}
+    Suppose that $G$ is a group such that $(ab)^i = a^i b^i$ for 3 consecutive
+    integers $i$, for all $a, b \in G$. Prove that $G$ is abelian.
+    % Topics in algebra, ch 2 ex 4
+\end{prob}
+
+\begin{prob}
+    Let $G$ be a nonempty finite set that is closed under an associative binary
+    operation such that for every $x, y, z \in G$, 
+    \begin{enumerate}
+        \item \textbf{(Left cancellation)} if $xy = xz$ then $y = z$, and;
+        \item \textbf{(Right cancellation)} if $yx = zx$ then $y =z$.
+    \end{enumerate}
+    Prove that $G$ is a group. Find an example that if one of the cancellation
+    laws were not assumed, that $G$ is not a group. (Find an example without left cancellation and without right cancellation)
+    % Note: Taken from topics in algebra, ch 2 ex 14
+\end{prob}
+
 
 \pagebreak
 \section{Subgroups}