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.

That does seem like the intuition you would like to capture by talking about integers as a group, maybe?

Kind of like how if you talk about points as elements of a vector space, it's useful to think of them as translations from the origin...

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.

Oh, you mean apple counting? Well, you have to shift those apples close together to recognize them as a group of things.

Right, so it's still visual, but it's not an example of the original suggestion to only define groups as groups of permutations.

It isn't?

> 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?

Not my bar, personally I think this is a crap attitude but one that I experienced.

It should be noted that group theory, as far as i can see, is a rather new development and a century is not much in the evolution of didactic.