So, I'm reading Introduction to Symmetry and Group Theory for Chemists to prepare for a project in Modern Algebra (and out of pure interest) and I've come across a problem that I thought would be pretty simple to solve. Clearly, I'm wrong and it's quite the brain teaser or I'm just not in the right mindset. The following is the problem.
Consider the symmetry group of an object for which the only covering operations are the identity and rotations by $120^\circ (=C_3)$ and $240^\circ (=C^2_3$) around the same axis. The multiplication table for this group is:
\begin{array}{c | c c c} \ & E & C_3 & C_3^2\\ \hline E & E & C_3 & C_3^2\\ C_3 & C_3 & C_3^2 & E\\ C_3^2 & C_3^2 & E & C_3\\ \end{array}
Can you determine all the possible sets of three complex numbers that satisfy the same multiplication table?
So, I've worked in my head a view numbers that could potentially work, but I keep running into a need to have a 4x4 table (i.e the $1,-1,i,-i$ symmetry group). I would imagine that in any case, the identity element should still be $0 + i^0$, but I could be wrong there. At this point, I would just appreciate a hint that might get me on the right track.
Thanks.
HINT: You've already (correctly) concluded that $E$ has to be $1$; so really the only question is, "What are the possible values for $C_3$?"
The properties a complex number $z$ must satisfy, in order to be a candidate for $C_3$, are:
$z\not=1$ (since $C_3\not=E$, and $E$ represents $1$);
$z^2\not=1$; and
$z^3=1$.
Now there are two approaches for finding all such $z$:
Brute force: express $z$ as $a+bi$, $a, b\in\mathbb{R}$, and solve $(a+bi)^3=1+0i$. Then check which of the solutions satisfy the other two constraints.
More cleverly: are you familiar with the polar representation of complex numbers, as $re^{i\theta}$? What does multiplication correspond to, for such numbers? This will give you a purely geometric solution.