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
Best Answer
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.