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Is there a $SL(2,\mathbb{Z})$-action on $\mathbb{Z}$?

I read this somewhere without proof and I am not sure if this is true.

Thank you for your help.

  • 0
    Sure, the trivial action. You may want to add some qualifiers to your question.2012-11-27
  • 1
    I am looking for a non-trivial action.2012-11-27
  • 1
    $\mathrm{SL}(2, \mathbb{Z}/2\mathbb{Z})$ is a quotient of the modular group, and its cardinal is 6. There aren't many groups of order 6, and in fact it's easily shown to be $\mathfrak{S}_3$, which can be mapped nontrivially to $\mathfrak{S}(\mathbb{Z})$. This gives you a nontrivial action of the modular group on $\mathbb{Z}$.2012-11-27
  • 0
    I was thinking that maybe you wanted an action with endomorphisms of $\mathbb{Z}$, but $\mathfrak{S}_3$ has $\mathbb{Z}/2\mathbb{Z}$ as a quotient, and this group acts nontrivially on $\mathbb{Z}$ by sending the generator on $-I$ (where $I$ is the identity), which is a group endomorphism.2012-11-27
  • 0
    Thank you for your help. I think I understand your first comment. But what do you mean by the second one. These seem not to be the same group actions which you describe.2012-11-27
  • 0
    An action on $\mathbb{Z}$ as what? A group? A ring?2012-11-27
  • 2
    Im a looking for an action on $\mathbb{Z}$ as a group.2012-11-28

2 Answers 2