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:


*

*The set is closed under the operation.

*The operation is associative.

*The exists and identity in the group.

*Each element in the group has an inverse which is also in the group.


Why does the operation need to be associative?
Thanks
 A: It is not that associativity is required for groups... That is quite backwards: the truth is actually that groups are associative.
Your question seems to come from the idea that people decided how to define groups and then began to study them and find them interesting. In reality, it happened the other way around: people had studied groups way before actually someone gave a definition. When a definition was agreed upon, people looked at the groups they had at hand and saw that they happened to be associative (and that that was a useful piece of information about them when working with them) so that got included in the definition.
If I may say so, it is this which is important to understand. The way we teach abstract algebra nowdays somewhat obscures this fact, but this is how essentially everything comes to be.  
A: In short, because that's how we choose to define them, because adding associativity allows us to study certain things more robustly.
There are algebraic structures that are group-like but don't satisfy all those axioms. A quasi-group need not be associative, and a loop need not be associative, but must have unity.
So, it may sound circular, but a group must be associative because if it is not, then it is not a group. A more apt question may be "why do we study group theory and not quasi-group theory?" In some senses, associativity gives us more freedom and power.
A: Groups are an abstraction. What do they abstract? The idea of symmetry. Symmetries are functions from a set to itself that preserve some structure of that set; for example, the symmetries of a square are rotations and reflections, and they preserve "squareness" (to put it vaguely). 
The multiplication in a group abstracts composition of symmetries (for example "rotate $90^{\circ}$, then reflect about the line $x = y$"), and composition of functions is always associative. 
A: The important thing to understand is that fulfilling these 4 axioms is all there is to being a group-in mathematics, we construct objects by imposing conditions on them and the underlying aggregates. (I'm not using the term "set" since mathematics doesn't always deal with sets,but you get the idea.)  What allows us to build a theory around a particuar defined object is the fact that whatever we choose the properties to be,those properties must be consistent with each other. It's the consequences of those properties cooexisting consistently that gives any object it's distinctive properties spelled out in the body of theorums and corollaries.  And that's the answer to your question: It turns out if you don't have associativity-that is, if axiom (2) is false, then axiom (4) is false because then you can have a set which has left inverses which are not right inverses and the definition assumes the inverse ( as well as the identity) is 2 sided. 
Consider an element x of a group G where there is a unique identity e and a left inverse l and a right inverse r for x. Then by axiom (4),they must be equal since both yield the unique identity and every inverse must be 2 sided. But: 
l = l*e = 1*( x*r) = (1*x)*r = e*r= r. 
But clearly the fact of l= r is dependent on the associativity of the operation. So whatever this algebraic structure is,it's not a group without it. 
Hope that answered your question.  
A: 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.
