TITLE: Are all algebraic commutative operations always associative?
QUESTION [16 upvotes]: I know that there are many algebraic associative operations which are commutative and which are not commutative.
for example multiplications of matrices as associative operation is not commutative.
I need to know about inverse of this!
I mean is there any algebraic commutative operation which is not associative?
can you show me sample?

REPLY [3 votes]: Extending the example in Rahul Narain's comment, if "$+$" is the addition operation in an Abelian group, then the operation $a\circ b=a+a+b+b$ is commutative but not associative if the group contains two elements $a$ and $b$ such that $a+a\not=b+b$ (so that $(a\circ 0)\circ b\not=a\circ(0\circ b)$).