Why is associativity required for groups? Why is associativity required for groups? I'm doing a linear algebra paper and we're focusing on groups at the moment, specifically proving whether something is or is not a group. There are four axioms: 1. The set is closed under the operation. 2. The operation is associative. 3. The exists and identity in the group. 4. Each element in the group has an inverse which is also in the group. Why does the operation need to be associative? Thanks

The formalist's answer is: it is just a definition. You could just as well consider studying algebraic structures that satisfy all the axioms for a group _except_ for associativity, and you would be then studying loops.

Now the question might be: why is the study of groups more ubiquitous than the study of loops? There are historical reasons (surely others with greater knowledge can expand upon this), and the fact that most loops that arise naturally when doing math are in fact groups is probably a reason too.