> What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? "Oh, curse this maths" - concludes the student (who, possibly, becomes the Minister for Science in the future).
> We get a totally different situation if we start off not with the group but with the concept of a transformation (a one-to-one mapping of a set onto itself) as it was historically. A collection of transformations of a set is called a group if along with any two transformations it contains the result of their consecutive application and an inverse transformation along with every transformation.
> This is all the definition there is. The so-called "axioms" are in fact just (obvious) properties of groups of transformations. What axiomatisators call "abstract groups" are just groups of transformations of various sets considered up to isomorphisms (which are one-to-one mappings preserving the operations). As Cayley proved, there are no "more abstract" groups in the world. So why do the algebraists keep on tormenting students with the abstract definition?
I agree that permutation groups are a good way of teaching. But there are also some disadvantages. Permutation groups have additional structure that groups lack. For example it makes sense to say that an element of a permutation group has no fixed points, but there could be an isomorphic group in which the corresponding element does have a fixed point. So students might try to use the idea of "has a fixed point" in a proof, even though this is incoherent.
Also, some groups aren't naturally transformation groups. Is it really easier to think of addition of integers as being composition of shifts?
I'm not sure I unterstand. Are you saying addition was not a natural transformation, in contrast to shifting?
I do have a Problem reasoning about binarry addition in terms of shift operations, but in the simplest case, a modulo 2 ring with reminder is just that. Easier? I don't know. Useful? Sure.
I mean that addition is a group operation, and that it's not easy to think of it as a transformation. The natural way of thinking about 2+2=4 is that two things and two more things make four altogether. No transfomations in sight.
Of course by Cayley's theorem you can think of this group as made by transformations. The element n is the transformation that "shifts" the numberline n steps in the positive direction. Then 2+2=4 means that shifting by two and then two again results in a shift of four. But this is a very forced way of thinking about addition.
Why do you need group theory at all when thinking about addition?
When you wonder "why does subtraction work" (guaranteed by the group structure), then thinking about shifts is a nice visual illustration, especially when you have modular arithmetic.
> So why do the algebraists keep on tormenting students with the abstract definition?
In my personal experience, the ability to extract the motivation and build up intuition is considered to be skill that the "students" have to develop for themselves and this is part of their learning process. Those that are not sufficiently successful will fail - and that is OK in such a system.
> It was hard to write, therefore it should be hard to read and this OK because why don't you go and write the comments never mind that I can't read my own code a months later. Fire and forget!
You'd prefer that a century of professional mathematicians of the past never have had their careers, since their way of thinking doesn't pass your bar of complexity?
to avoid duplicating the string combinations? The conditional is unfortunate, would like to get rid of it. But should be straightforward to extend if we want to introduce further string combinations for numbers divisible by, e.g., 7. Though anticipating such things goes against the spirit of k I suppose. :-)
> What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? "Oh, curse this maths" - concludes the student (who, possibly, becomes the Minister for Science in the future).
> We get a totally different situation if we start off not with the group but with the concept of a transformation (a one-to-one mapping of a set onto itself) as it was historically. A collection of transformations of a set is called a group if along with any two transformations it contains the result of their consecutive application and an inverse transformation along with every transformation.
> This is all the definition there is. The so-called "axioms" are in fact just (obvious) properties of groups of transformations. What axiomatisators call "abstract groups" are just groups of transformations of various sets considered up to isomorphisms (which are one-to-one mappings preserving the operations). As Cayley proved, there are no "more abstract" groups in the world. So why do the algebraists keep on tormenting students with the abstract definition?