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Math Help - Elements of quotient groups

  1. #1
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    Elements of quotient groups

    Hello,
    I have a group G=(\mathbb{R}^{2},+) and it's subgroup H=\{{(x,y) \in \mathbb{R}^{2}: x+3y=0\}. What are the elements of quotient group G / H? How to find them?
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Elements of quotient groups

    Quote Originally Posted by MachinePL1993 View Post
    Hello,
    I have a group G=(\mathbb{R}^{2},+) and it's subgroup H=\{{(x,y) \in \mathbb{R}^{2}: x+3y=0\}. What are the elements of quotient group G / H? How to find them?
    Hi MachinePL1993!

    The elements of a quotient group are cosets, which are of the form (a + H) for an additive group.

    In your case a is point in \mathbb{R}^{2} and H is your line.
    When we add a point to a line, we get a parallel line.
    So G/H consists of all lines parallel to H.
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