Undergrad What are double cosets in group theory?

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SUMMARY

This discussion focuses on the concept of double cosets in group theory, particularly in the context of the notation used in mathematical papers. The notation H\G/K denotes a double coset, where the backslash represents left cosets and the slash represents right cosets. The participants clarify that the distinction between left and right operations is crucial for understanding double cosets. The conversation references a specific paper from arXiv, which provides an example involving the groups ∆1\SU(3)/∆1 and ∆\U(3)/∆.

PREREQUISITES
  • Understanding of group theory concepts, specifically cosets
  • Familiarity with notation in mathematical literature
  • Knowledge of left and right group operations
  • Basic comprehension of quotient groups
NEXT STEPS
  • Study the properties of double cosets in group theory
  • Explore the implications of left and right cosets in algebraic structures
  • Read the referenced paper on arXiv for practical examples
  • Investigate the role of associativity in group operations
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in advanced group theory concepts, particularly those studying cosets and their applications.

Heidi
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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
 
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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?
 
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?
 
Heidi said:
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.
 
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.
 

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