Groups

Discipline summary

Start with a set and an operation acting on pairs formed from that set.
0) Assume every possible pair yields something from that operation.
1) Assume you can chain that operation without taking care in which order that operation is performed.
2) Assume there exists an element which leaves any other element invariant under that operation.
3) Assume for each element there exists an inverse: Another unique element, such that the operation applied to them in any order will result in the element leaving others invariant.

you get a structure composed of the set and a 2-parameter function representing the operation(0). That function or operation has a neutral element(2) and is associative(1). Additionally the function can be inverted in each parameter: Applying the operation to that parameter's inverse object(3) and the given output. Of course, applied in the same order as was the order of the parameters.

That's a group! It's called commutative when the order of the operator doesn't change the result. Groups can be thought of as sets of matrices, along with matrix multiplication. Another interesting point of view is bijective functions on some set of numbers...

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