Understanding Coset Representation and its Role in Group Theory

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Discussion Overview

The discussion revolves around the concept of coset representation in group theory, specifically focusing on the definition and implications of coset representatives, particularly in the context of normal and non-normal subgroups. The scope includes theoretical aspects and conceptual clarifications related to group theory.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether a "coset representation" exists and suggests that it may refer to an element that factors outside the coset.
  • Another participant clarifies that the standard representation of a left coset is "aH," where "a" is a member of group G and "H" is a subset of G.
  • A different participant notes that a coset representative would be the "a" in "aH" but raises concerns about the terminology when "H" is not a normal subgroup, suggesting that uniqueness of representatives could complicate the discussion.
  • There is a reiteration of the standard representation of a left coset, confirming the earlier point about "aH."

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of discussing coset representatives when the subgroup is not normal, indicating a lack of consensus on the implications of this distinction.

Contextual Notes

Participants highlight that the uniqueness of coset representatives can be problematic when "H" is not a normal subgroup, which may affect the calculations and interpretations within the context of quotient groups.

pivoxa15
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Is there a thing called a coset representation?

If so is it the element that factors outside the coset?

i.e. {4,8,12,16,...} = 4{1,2,3,4,...} so 4 is the coset representation for {4,8,12,16,...}
 
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The standard way of representing a (left)coset is as "aH" where a is a member of group G and H is a subset of G. Is that what you are talking about?
 
I believe a coset representative would be the a in aH. Of course, talking about "coset representatives" when H isn't a normal subgroup is a little odd. For one, if H isn't a subgroup, the "representative" might be unique, and that's bad for most things you want to do with cosets (like in your example, if H = {1, 2, 3, 4...}, then 4 is a "coset representative" for {4, 8, 12, 16...}, but it's the only one, assuming the pattern continues as indicated). Also, you really only talk about coset representatives in terms of calculations in the quotient group, and that only makes sense if H is normal.
 
HallsofIvy said:
The standard way of representing a (left)coset is as "aH" where a is a member of group G and H is a subset of G. Is that what you are talking about?

Yes.
 

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