Possibly ambiguous Cayley table construction If $\{a,b\}$ with operation $*$ is to be a group, with $a$ the identity element, then what must the Cayley table be? 
I thought the following:
\begin{array}{c|cc}
* & a & b\\
\hline a & a & b\\
b& b& ?
\end{array}
Can the question mark be either $a$ or $b$? Or does it have to be $a$? The information given to me does not seem to make it clear that it has to be one or the other. 

Every element in a group has an inverse. In your case, there must be some element $x \in \{a,b\}$ for which $b * x = a$.
To see this another way, note that no two elements in a row are equal. This is because if $b * x = b * y$, then by multiplying on the left by $b^{-1}$, we have $x = y$.
