What are double cosets in group theory?

[Heidi]

Hi Pfs
It is the first time that reas something about "double cosets"
it was in this paper
https://arxiv.org/pdf/0810.2091.pdf
At page 4 i read
∆1\SU(3)/∆1 = ∆\U(3)/∆
Could you help to understand what are these sets (or cosets)?
thanks

 

[martinbn]

Do you know what a coset (double coset) is? In other words are you asking what double cosets are, or are you asking what these specific ones are?

 

[Heidi]

I would like to get the meaning of the slashs and anti slashs
in the wikipedia double coset article i found the notations
H\G/K and (H\G)/K and H\(G/K)
i suppose that the slash is for a quotient but what is the antislash for?

 

[fresh_42]

i suppose that the slash is for a quotient but what is the antislash for?

If we have double cosets, then we have to distinguish right and left operations, i.e. multiplication from left and right. The two slashes are meant to do this, backslash for the left coset, and slash for the right coset. There is no deeper meaning.

 

[WWGD]

Ah, I got confused until I realized the groups by which we fsctor are all subgroups of G. Then also associaivity kicks in and the triple product on both sides makes sense.