Understanding Coset Representation and its Role in Group Theory

[pivoxa15]

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,...}

 

[HallsofIvy]

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?

 

[Mystic998]

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.

 

[pivoxa15]

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.