Group theory originated in a very visual and geometric ways studying the symmetries of various objects, before the modern definition of group was given. Many groups studied today are still geometrical in nature: the dihedral groups are the groups of symmetries of regular polygons, the symmetric groups are the groups of permutations of a set with N elements and so on and so forth.
After the notion of group was formalized (a set with an operation satisfying some axioms) it was natural to ask oneself what was the advantage of this apparently more general and abstract definition. Turns out (that's a theorem of Cayley) that every group is isomorphic to a subgroup of a symmetric group (that is a group of permutations on some set) so arguably we didn't get that much more generality while moving from the first geometric groups to the modern abstract ones!
> 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?
I don't see how a permutation group lends itself well to a geometric or visual interpretation in general. From my understanding, a permutation is just a one-to-one and onto function from a set to itself. For a set of N elements there are N! possible permutations, any subset of which could be used to generate a permutation group. How do you propose to interpret all of these groups geometrically, as opposed to abstractly?
> How do you propose to interpret all of these groups geometrically,
How do you not? This pure maths wank really grinds my gears. Of course you can substitute other perceptive phenomena next to the visual. But even your abstract thoughts have to be accomodated in the geometry of the brain, if you think about it.
>> one-to-one and onto funktion
>
> I don't think that's the established terminology.
It is. Injective and surjective are more "international", but those English names are fine as well. What's missing is that the function sshould map a set onto itself.
So every finite group is isomorphic to a subgroup of a N dimensional simplex? And then we should build our intuition about these groups via a geometric visualization? These are not rhetorical questions, I'm currently learning abstract algebra and trying to understand.
I agree that there are many natural groups that should be given much more attention when teaching group theory.
But the virtue of abstractness to me is in the notion of isomorphism, which is surely critical to how we understand groups. Then you can find same (that is, isomorphic) groups with different representations - like sitting as subgroups in two different symmetric groups.
And to define when two groups are isomorphic is more or less the same as to define what a group is abstractly.
> Group theory originated in a very visual and geometric ways studying the symmetries of various objects
Did it? As far as I know, group theory started as method to describe the possible permutations of roots of polynomials, so it was very abstract right from the beginning.
And while there are certain geometric interpretation of some groups and the ways these groups act on objects, the "visual" part stops as soon as the groups gets more general or have more structure.
You are correct, group theory did start with the permutations of roots of polynomials. In the other hand, the modern perspective is to consider these groups as coming from geometry (the etale fundamental group point of view) and is really a very successful theory.
I think it is generally useful to try and give a geometric interpretation to whatever we are interested in and groups certainly benefit from such an approach.
Moreover, the groups relevant to physics are very geometric things (they are literally spaces in the sense that the real number line is a space ) and the geometry plays a large role in these groups.
In fact I am struggling to think of an application of groups to mathematics where geometry does not come into play.
Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory).
> In fact I am struggling to think of an application of groups to mathematics where geometry does not come into play.
Number theory is another example, finding roots of polynomials was already mentioned.
> Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory).
It connects linera algera to group theory, yes. But representation theory is by far the least "visual" branch of group theory I can think of. It is on a similar "abstract nonsense" level as category theory (and I don't mean that in a bad way, it is just how things are). The only visualizations I can find in my representation theory scripts are commutative diagrams between group homomorphisms, vector space homomorphims and some module endomorphisms and what not :)
There's certainly a question of appropriate pedagogy. In my own experience though, I find the abstract definitions easier to apply in novel situations rather than specific representations.
If I see a thing in the wild that has invertable operations, then I immediately think about an underlying group structure. As subgroups of Sn though, it's harder for me to make that connection.
When learning abstract algebra and category theory and such, thinking up several concrete examples on my own pretty much dissolves the "arbitrary list of axioms" feel for me and makes the definitions seem quite natural.
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[ 3.4 ms ] story [ 73.4 ms ] threadAfter the notion of group was formalized (a set with an operation satisfying some axioms) it was natural to ask oneself what was the advantage of this apparently more general and abstract definition. Turns out (that's a theorem of Cayley) that every group is isomorphic to a subgroup of a symmetric group (that is a group of permutations on some set) so arguably we didn't get that much more generality while moving from the first geometric groups to the modern abstract ones!
> 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?
Also, some groups aren't naturally transformation groups. Is it really easier to think of addition of integers as being composition of shifts?
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.
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.
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...
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.
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.
I don't think that's the established terminology.
> How do you propose to interpret all of these groups geometrically,
How do you not? This pure maths wank really grinds my gears. Of course you can substitute other perceptive phenomena next to the visual. But even your abstract thoughts have to be accomodated in the geometry of the brain, if you think about it.
It is. Injective and surjective are more "international", but those English names are fine as well. What's missing is that the function sshould map a set onto itself.
Thanks! one-to-one and onto is the terminology my American courses are using.
> What's missing is that the function sshould map a set onto itself.
Just for the record I initially stated "a permutation is just a one-to-one and onto function from a set to itself" as you say it must.
But the virtue of abstractness to me is in the notion of isomorphism, which is surely critical to how we understand groups. Then you can find same (that is, isomorphic) groups with different representations - like sitting as subgroups in two different symmetric groups.
And to define when two groups are isomorphic is more or less the same as to define what a group is abstractly.
Did it? As far as I know, group theory started as method to describe the possible permutations of roots of polynomials, so it was very abstract right from the beginning.
And while there are certain geometric interpretation of some groups and the ways these groups act on objects, the "visual" part stops as soon as the groups gets more general or have more structure.
I think it is generally useful to try and give a geometric interpretation to whatever we are interested in and groups certainly benefit from such an approach.
Moreover, the groups relevant to physics are very geometric things (they are literally spaces in the sense that the real number line is a space ) and the geometry plays a large role in these groups.
In fact I am struggling to think of an application of groups to mathematics where geometry does not come into play.
Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory).
Number theory is another example, finding roots of polynomials was already mentioned.
> Even the abstract classification of finite simple groups uses ideas from geometry on a very crucial way (representation theory).
It connects linera algera to group theory, yes. But representation theory is by far the least "visual" branch of group theory I can think of. It is on a similar "abstract nonsense" level as category theory (and I don't mean that in a bad way, it is just how things are). The only visualizations I can find in my representation theory scripts are commutative diagrams between group homomorphisms, vector space homomorphims and some module endomorphisms and what not :)
If I see a thing in the wild that has invertable operations, then I immediately think about an underlying group structure. As subgroups of Sn though, it's harder for me to make that connection.
When learning abstract algebra and category theory and such, thinking up several concrete examples on my own pretty much dissolves the "arbitrary list of axioms" feel for me and makes the definitions seem quite natural.
I am downloading all of his channel right now to watch and I don't know how to thank you :)