The following multiplication table was given to me as a class exercise. I should have all the necessary information to fill it completely in. However, I'm not sure how to take advantage of the relations I am given to fill it in?
The Question
A group has four elements $a,b,c$ and $d$, subject to the rules $ca = a$ and $d^2 = a$. Fill in the entire multiplication table.
\begin{array}{c|cccc} \cdot & a & b & c & d \\ \hline a& & & & \\ b& & & & \\ c& a & & & \\ d& & & & a \end{array}
I imagine I might proceed like this:
To find $ab$, write $a = d^2$ and thus $ab = d^2b = db\cdot b$....but my chain of reasoning always stops around here.