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Rubik's cube group element with the smallest order

  1. Jan 16, 2015 #1
    Wikipedia says that largest order of any element of Rubik's cube group is 1260 [PLAIN]http://upload.wikimedia.org/math/e/1/c/e1cff178a2562422492a4140a38f93ff.png. [Broken] http://en.wikipedia.org/wiki/Rubik's_Cube_group
    What about element of smallest order (except the identity element)? I'll appreciate any example with small order.
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Jan 16, 2015 #2
    How about R2, or R2L2U2D2F2B2
    If you have an element G of order n, you can easily find an element with an order of any divisor d of n
    [tex] G^{n/d} [/tex]
     
  4. Jan 16, 2015 #3

    jbunniii

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    Twisting any of the faces 180 degrees will have order 2, the smallest possible non-identity order.

    [edit] @willem2 beat me to it :D
     
  5. Jan 16, 2015 #4
    Thank you. These are kind of uninteresting. Any interesting examples ... :)
     
  6. Jan 16, 2015 #5
    Is it easy to find? For example 10 divides the 1260 (order of e1cff178a2562422492a4140a38f93ff.png ). How I am going to find an element of order 10 from this ... ??
     
  7. Jan 16, 2015 #6

    jbunniii

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    Apply ##(RU^2D^{-1}BD^{-1})## 1260/10 = 126 times. It will be good exercise for your wrists. :-)
     
  8. Jan 16, 2015 #7
    I am not going to do that :)
     
  9. Jan 16, 2015 #8
    But how I'll convert that position to L R U D notation?
     
  10. Jan 16, 2015 #9
    google for "rubiks cube solver"
     
  11. Jan 17, 2015 #10
    Thank you very much. Now I know.
     
  12. Jan 17, 2015 #11
    This was actually a serious reply. Solvers that can find an optimum solution exist.
    I downloaded the solver from http://kociemba.org/cube.htm and pasted in RU2D'BD' 126 times, and it immediately found D' R D2 R' D2 R F D2 F' D' R' D2
    to generate the same pattern
     
  13. Jan 18, 2015 #12
    I know. And I found it extremely helpful. What I meant by "Now I know" is that, I can find out notation for any position by simply solving it by the cube solver. Actually it was the best thing that happened on the day. Thanks. :)
     
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