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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
by OscarCunningham 9y ago
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?
- ndnjjfhshhvjj 9y agoI'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.
- OscarCunningham 9y agoI 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.
- posterboy 9y agoOh, you mean apple counting? Well, you have to shift those apples close together to recognize them as a group of things.
- OscarCunningham 9y agoRight, so it's still visual, but it's not an example of the original suggestion to only define groups as groups of permutations.
- posterboy 9y agoIt isn't?
- mbrock 9y agoThat 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...
- gowld 9y agoWhy 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.