The intuition for why there should exist a correspondence between the cosets of the stabilizer and the elements in the orbit is done very well, thanks.
Surjective means that all elements in O_a will have an element mapped onto it. O_a are all elements of the form g.a for all g in G since our domain is all of G (just partitioned into cosets) we can just take the element that acted on a (or even its entire coset as the result is identical) to produce the element in O_a. Say an element in O_a was produced by b.a we can just let b or any member of its coset act on a to get b.a back again.
All elements in G create the entire orbit. There is nothing missing from the orbit. If there was (call it a*), we know that a* was produced from some element g in G and a in A, by definition of an orbit. You map a* to the coset with g, proving surjection.
The intuition for why there should exist a correspondence between the cosets of the stabilizer and the elements in the orbit is done very well, thanks.
This is gold 🙌
Amazing. Thank you very much.
amazing videos I keep coming back to them
a make over removing obvious ambiguities in notation would make an excellent model for this theorem
How would you show the correspondence between the left cosets of G_a and the elements in O_a is surjective?
Surjective means that all elements in O_a will have an element mapped onto it. O_a are all elements of the form g.a for all g in G since our domain is all of G (just partitioned into cosets) we can just take the element that acted on a (or even its entire coset as the result is identical) to produce the element in O_a. Say an element in O_a was produced by b.a we can just let b or any member of its coset act on a to get b.a back again.
18:56 why can't we use the right inverse of a, to show that g and g bar are equal? Is that not valid?
if Oa1={a1,a2 ....am} Is Oa2 also the same as Oa1?
Yes. He proved this in the first part.
Why is it bijective? I understand it's injective, but why is it surjective?
All elements in G create the entire orbit. There is nothing missing from the orbit. If there was (call it a*), we know that a* was produced from some element g in G and a in A, by definition of an orbit. You map a* to the coset with g, proving surjection.