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jango_IN_DTOWN

  • one year ago

Is the group of all rotations of R^2 cyclic?

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  1. jango_IN_DTOWN
    • one year ago
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    @ganeshie8

  2. jango_IN_DTOWN
    • one year ago
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    @nincompoop

  3. ganeshie8
    • one year ago
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    check http://math.stackexchange.com/questions/148111/proof-that-finite-group-of-rotations-of-plane-is-cyclic

  4. jango_IN_DTOWN
    • one year ago
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    Its not showing any solution or any hint..

  5. ganeshie8
    • one year ago
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    let me just give you the proof : https://s3.amazonaws.com/upload.screenshot.co/73cd8e1818

  6. jango_IN_DTOWN
    • one year ago
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    What is the case when the group becomes infinite? In the question the finite or infinite case was not mentioned...

  7. ganeshie8
    • one year ago
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    I think we must have a "minimum positive angle of rotation" in the group for it to be cyclic; for infinite groups, we cannot guarantee the existence of a minimum rotation element in the group. so, my hunch is that there may exist some infinite groups of plane rotations that are not cyclic

  8. ganeshie8
    • one year ago
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    for example, consider the group of rotations with angles whose measures are real numbers then, its easy to see, since there is no minimum positive element, the previous proof using euclid theorem fails

  9. jango_IN_DTOWN
    • one year ago
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    So in general it is not cyclic, we may conclude?

  10. ganeshie8
    • one year ago
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    On the other hand, the infinite group of rotations with angles whose measures are "integers" is cyclic. because there exists a minimum positive rotation angle : "1" those two examples of infinite groups make me believe that some infinite groups of plane rotations are cyclic and some are not.

  11. ganeshie8
    • one year ago
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    Yes, not necessarily cyclic...

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