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Which lines in $\mathbb R^2$ define a subgroup?

I know that the line $y=x$ in $\mathbb{R}^2$ gives a subgroup, but I can't figure out the other ones.

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    $\begingroup$ HINT: Every vector subspace of $\Bbb R^2$ is a subgroup of $\Bbb R^2$. $\endgroup$ Nov 23, 2015 at 3:45
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    $\begingroup$ Hint: a subgroup must contain the identity. $\endgroup$ Nov 23, 2015 at 3:45
  • $\begingroup$ see this $\endgroup$
    – Bumblebee
    Nov 23, 2015 at 4:06
  • $\begingroup$ So all the straight lines that pass through the origin are subgroups? $\endgroup$
    – MilTom
    Nov 23, 2015 at 4:14
  • $\begingroup$ Correct. The line must pass through the origin, as $(0,0)$ is the identity of $\mathbb{R}^2$. Now just verify that any such line is a subgroup. $\endgroup$ Nov 23, 2015 at 4:25

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