Group (Associativity of Binary Operators)

[jeff1evesque]

Statement:
One of the key elements in being in what people call group is that elements must be associative.

So this means if we take any three elements from what we propose to be a group, they should be associative,
a*(b*C) = (a*b)*c

Question:
Suppose we do have a group with elements a, b, c.
Would the following be true,
a*(b*C) = (a*b)*c = (c*a)*b

 

[CompuChip]

It would be true if and only if c commutes with ab. In that case, you only need associativity in a(bc) = (ab)c = c(ab) = (ca)b.
However, in general that is not true. Probably your favorite non-abelian group provides an easy counterexample.