Does the requirement that the group be a set (as opposed to a proper class) have a meaningful role in this early discussion? The set requirement seems to impact studies of equivalence classes of group elements under the orbits imposed by a subgroup, which is one of the principle tools in the classification of groups.
One equivalent way to define what a group is is that it is a concrete category with only one object, and such that every morphism is an isomorphism. A very obvious generalization would be to remove the need for the category to be concrete.
However, while I know a multitude of non-concrete categories, I can't say that I know of many non-concrete groups. I suppose that the surreals count, although you probably want the full structure of an ordered field (generalized to allow for proper classes), rather than just thinking about the group structure.
In any case, I'm not planning on getting into the set-theoretical weeds. I used the word "set", but I could have used the word "collection" pretty much just as well. But since "set" has a formal definition and it is what is normally used, I think that it is appropriate.
If at any point in this series you decide to mention at all "generalizations" of groups, i.e. magmas, semigroups, loops (not necessarily derived as generalizations, of course, only using the word to say that the axioms are loosened), I do think it would be nice to at least brush over some of those set-theoretical weeds. Even if it's just a comment on where to find more, if interested :O
In addition to the obvious ① additive and ② multiplicative group-like classes of surreal numbers and their obvious proper sub-classes which are closed under the group-like operation, there is ③ the Grothendieck natural addition group (an equivalence class on the natural addition on pairs of ordinals) which is isomorphic to a subclass of the integer-like surreals in that no n and m exist such that m = n √2, I can think of ④ 2^On which is all subclasses of the Ordinal numbers under symmetric difference of classes as easy examples of group-like proper classes.
Proving that a group-like class has a particular structure or even that it is group-like may run into problems for groups complex enough so that we run out of naming schemes for the different subclasses of elements.
I've given exams solving problems in groups rings and fields in high-school. But nobody not even the teachers explained what these structures have to do with solving algebraic equations. I found out later that it has to do with symmetry. But how ? Everybody drop the words symmetry of solutions and that's it , they don't do deeper. What has the symmetry of solutions in a descartian reference system to do with solubility of the equation? Maybe you'll write a detail article. I'll follow your substack.
You mean like in Galois theory, showing that the quintic isn't solvable in radicals? Yes, I do plan to cover that, but it won't be as a single article---it will be part of a long series of lectures on Galois theory. It's really not something you can explain quickly.
There's an easier application to finding solutions to partial differential equations by looking for symmetries. This is still quite deep, but I can at least give some basic flavor of what it looks like.
Yes thank you! Please start with the beginning. Why the solution of ax=0 forms a group. Then why the solution of ax2+bx+c=0 form a group. From the particular to the general. Ending in the quintic. I subscribed.
It's not the solutions that form a group. It's the collection of field automorphisms that form a group, but they happen to shuffle around the roots of any polynomial.
Does the requirement that the group be a set (as opposed to a proper class) have a meaningful role in this early discussion? The set requirement seems to impact studies of equivalence classes of group elements under the orbits imposed by a subgroup, which is one of the principle tools in the classification of groups.
One equivalent way to define what a group is is that it is a concrete category with only one object, and such that every morphism is an isomorphism. A very obvious generalization would be to remove the need for the category to be concrete.
However, while I know a multitude of non-concrete categories, I can't say that I know of many non-concrete groups. I suppose that the surreals count, although you probably want the full structure of an ordered field (generalized to allow for proper classes), rather than just thinking about the group structure.
In any case, I'm not planning on getting into the set-theoretical weeds. I used the word "set", but I could have used the word "collection" pretty much just as well. But since "set" has a formal definition and it is what is normally used, I think that it is appropriate.
If at any point in this series you decide to mention at all "generalizations" of groups, i.e. magmas, semigroups, loops (not necessarily derived as generalizations, of course, only using the word to say that the axioms are loosened), I do think it would be nice to at least brush over some of those set-theoretical weeds. Even if it's just a comment on where to find more, if interested :O
In addition to the obvious ① additive and ② multiplicative group-like classes of surreal numbers and their obvious proper sub-classes which are closed under the group-like operation, there is ③ the Grothendieck natural addition group (an equivalence class on the natural addition on pairs of ordinals) which is isomorphic to a subclass of the integer-like surreals in that no n and m exist such that m = n √2, I can think of ④ 2^On which is all subclasses of the Ordinal numbers under symmetric difference of classes as easy examples of group-like proper classes.
Proving that a group-like class has a particular structure or even that it is group-like may run into problems for groups complex enough so that we run out of naming schemes for the different subclasses of elements.
That's all I've got. Thank you for your patience.
I've given exams solving problems in groups rings and fields in high-school. But nobody not even the teachers explained what these structures have to do with solving algebraic equations. I found out later that it has to do with symmetry. But how ? Everybody drop the words symmetry of solutions and that's it , they don't do deeper. What has the symmetry of solutions in a descartian reference system to do with solubility of the equation? Maybe you'll write a detail article. I'll follow your substack.
You mean like in Galois theory, showing that the quintic isn't solvable in radicals? Yes, I do plan to cover that, but it won't be as a single article---it will be part of a long series of lectures on Galois theory. It's really not something you can explain quickly.
There's an easier application to finding solutions to partial differential equations by looking for symmetries. This is still quite deep, but I can at least give some basic flavor of what it looks like.
Yes thank you! Please start with the beginning. Why the solution of ax=0 forms a group. Then why the solution of ax2+bx+c=0 form a group. From the particular to the general. Ending in the quintic. I subscribed.
It's not the solutions that form a group. It's the collection of field automorphisms that form a group, but they happen to shuffle around the roots of any polynomial.
But how are groups?