Chapter 3.1, Problem 9E

In Exercises $1-14$, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition $3.1$ that fails to hold.

The set of all complex numbers $x$ that have absolute value $1$, with operation multiplication. Recall that the absolute value of a complex number $x$ written in the form $x=a+bi$, with $a$ and $b$ real, is given by $\left|x\right|=\left|a+bi\right|=\sqrt{a^2+b^2}$.