TITLE: About group multiplication table
QUESTION [6 upvotes]: The following multiplication table was given to me as a class exercise.
The Question
A group has five elements $a$, $b$, $c$, $d$ and $e$, subject to the rules $ab=d$, $ca=e$ and $dc=b$. Fill in the entire multiplication table. 
$\begin{array}{c|ccccc}
\cdot & a & b & c & d & e \\ 
\hline
a &  &  &  &  & \\ 
b &  &  &  &  & \\ 
c &  &  &  &  & \\ 
d &  &  &  &  & \\
e &  &  &  &  & 
\end{array}$
I'm not so sure how to solve this. Any help?

REPLY [2 votes]: Now, I think that the answer is 
$\begin{array}{c|ccccc}
\cdot & a & b & c & d & e \\ 
\hline
a & b & d & e & c & a \\ 
b & d & c & a & e & b \\ 
c & e & a & d & b & c \\ 
d & c & e & b & a & d \\
e & a & b & c & d & e
\end{array}$
Is this true?