Is it possible to prove associativity from $a(bc)=a(cb)$? With the given information, I'm trying to prove that an operation is both associative and commutative. 
Let $(A,*)$ be a set and a binary operation such that for all $a,b,c \in A$
$$a*(b*c) = a*(c*b),$$
and
for some $e \in A$ and for any $a \in A$ $$e*a = a*e = a.$$
So far, proving that $*$ is commutative has been easy, but finding a method to prove associativity has escaped me. I don't want the solution to the problem, but could anyone help in assuring me that associativity is even possible with the given assumptions.
Thanks!

No.
Counterexample:
$A = \{a,b,c \}$
\begin{array}{c|ccc}
*&a&b&c\\
\hline
a&a&b&c\\
b&b&c&c\\
c&c&c&b\\
\end{array}
$*$ is clearly commutative (and hence also $a * (b * c) = a * (c * b)$), $a$ is its identity element, but $*$ is not associative:
$b*(b*c)=b*c=c$ but $(b*b)*c=c*c=b$
