# Coset representation?

1. Jan 31, 2008

### 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,....}

2. Jan 31, 2008

### 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?

3. Jan 31, 2008

### 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.

4. Jan 31, 2008

Yes.