# Thread: Groups - Cosets of specific groups

1. ## Groups - Cosets of specific groups

Let G be $\displaystyle R^2$ , H={$\displaystyle (t,3t) : t\in R$}.

I need to find the left cosets of G for the subgroup H. (describe it)

For every 't' I choose, there's still many options left, and there's not any 'bunch' of t's I can choose that will cover all the options (options=G, when I think of it).

How can I still explain this, in a mathematical way? Or, IS there any way to define the cosets?

Thanks

Let G be $\displaystyle R^2$ , H={$\displaystyle (t,3t) : t\in R$}.

I need to find the left cosets of G for the subgroup H. (describe it)

For every 't' I choose, there's still many options left, and there's not any 'bunch' of t's I can choose that will cover all the options (options=G, when I think of it).

How can I still explain this, in a mathematical way? Or, IS there any way to define the cosets?
Think of this geometrically. The set $\displaystyle G=R^2$ represents two-dimensional space, and the set of points $\displaystyle H=\{(t,3t) : t\in R\}$ represents a line through the origin with gradient 3. The cosets of H are all the lines parallel to that one. So for every real number k, the set $\displaystyle \{(t,3t+k) : t\in R\}$ will be a coset (it represents the line parallel to H with y-intercept k), and those are all the cosets of H.

3. Awww!

I see it now! It's beautiful, thanks !!