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Math Help - Groups - Cosets of specific groups

  1. #1
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    Groups - Cosets of specific groups

    Let G be R^2 , H={ (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
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  2. #2
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    Quote Originally Posted by adam63 View Post
    Let G be R^2 , H={ (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 G=R^2 represents two-dimensional space, and the set of points 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 \{(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.
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  3. #3
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    Awww!

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