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
Solution 1:
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.
Solution 2:
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.
Solution 3:
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.