<s>[INST] complete the table by providing an example of a binary operation $*$ defined on $\{a , b ,c\}$ I have a problem with one of my questions. The question is:
complete the table by providing an example of a binary operation $*$ defined on $\{a , b ,c\}$
such that $*$ is commutative and has the identity element $c$.
\begin{array}{c|ccc}
\ast & a & b & c \\ 
\hline
a & a & ? & ? \\ 
b & ? & b & ? \\ 
c & ? & ? & c \\ 
\end{array}
(I need the letters in place of the ?)
I don't understand what they want me to do ???
 [/INST] For $a$ to be the identity element, it must be $a\ast b=b=b\ast a$ and $a\ast c=c=c\ast a$, so you have:
\begin{array}{c|ccc}
\ast & a & b & c \\ 
\hline
a & a & \color{lime}{b} & \color{lime}{c}\\ 
b & \color{lime}{b} & b & ? \\ 
c & \color{lime}{c} & ? & c \\ 
\end{array}
Then, for commutativity to hold, this is for the Cayley table to be symmetric, you have only three choices:
\begin{array}{c|ccc}
\ast & a & b & c \\ 
\hline
a & a & b & c\\ 
b & b & b & \color{red}{a} \\ 
c & c & \color{red}{a} & c \\ 
\end{array}
\begin{array}{c|ccc}
\ast & a & b & c \\ 
\hline
a & a & b & c\\ 
b & b & b & \color{red}{b} \\ 
c & c & \color{red}{b} & c \\ 
\end{array}
\begin{array}{c|ccc}
\ast & a & b & c \\ 
\hline
a & a & b & c\\ 
b & b & b & \color{red}{c} \\ 
c & c & \color{red}{c} & c \\ 
\end{array}
Exactly at the same way, you have three choices if $b$ is the identity, and three more if the identity is $c$. In this last case:
\begin{array}{c|ccc}
\ast & a & b & c \\ 
\hline
a & a & ? & \color{lime}{a}\\ 
b & ? & b & \color{lime}{b} \\ 
c & \color{lime}{a} & \color{lime}{b} & c \\ 
\end{array}
and so on, as above.
</s>