In C$3$v both C$.,$ and C~ belong to the sa me class, listed as $2$C$3$ in the character

tabie. As members of the sa me class their characters for any representation are

the same.

$($a$)$ Demonstrate that both C$3$ and C~ have a character of $1$ for the A $.$ I

representation, by which z transforms.

$($b$)$ Demonstrate that both C$3$ and C~ have a character of $-1$ for the E

representation, by which x and y transform degenerately. $[$Hint: Write the

$2$ x $2$ translorrnation matrices describing the actions of the operations on a

point $($x$,$ y$)$ and determine their characters.$]$