1. (15 points) The following is an incomplete Cayley table for a group \( G=\{a, b, c\} \).
\begin{tabular}{|l||l|l|l|}
\hline\( * \) & \( a \) & \( b \) & \( c \) \\
\hline \hline\( a \) & \( c \) & & \\
\hline\( b \) & & \( b \) & \\
\hline\( c \) & & & \( a \) \\
\hline
\end{tabular}
So from the table, we know \( a * a=c, b * b=b, c * c=a \).
(a) What is the identity? Show your reasoning.
(b) Complete the table.
(c) What is the inverse for each element \( a, b, c \)
(d) Find the orders \( |a|,|b| \) and \( |c| \).
(e) Is \( G \) a cyclic group? If yes, list all generators.