TITLE: When I state $(a\cdot b) \cdot c = (a \cdot c) \cdot b$ , what properties am I using?
QUESTION [7 upvotes]: When I state $(a\cdot b) \cdot c = (a \cdot c) \cdot b$ , am I using only the commutative property or both the commutative and associative property?

REPLY [5 votes]: So far nobody has given counterexamples showing that associativity alone or commutativity alone are not sufficient to guarantee $(a\cdot b)\cdot c=(a\cdot c)\cdot b$.
Multiplication of $(n\times n)$-matrices is certainly associative, but not commutative when $n\geq2$. Choose any two non-commuting $(2\times2)$-matrices $b$ and $c$, and let $a$ be the $(2\times2)$ identity matrix. Then $(a\cdot b)\cdot c\ne(a\cdot c)\cdot b$.
For complex numbers $z$, $w$ define
$$z\ *\ w:=\overline{z\cdot w}\ .$$
This product is certainly commutative, but $(1*1)*i\ne(1*i)*1$.