# Rubik's cube group element with the smallest order

## Main Question or Discussion Point

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.

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What about element of smallest order (except the identity element)? I'll appreciate any
If you have an element G of order n, you can easily find an element with an order of any divisor d of n
$$G^{n/d}$$

jbunniii
Homework Helper
Gold Member
Twisting any of the faces 180 degrees will have order 2, the smallest possible non-identity order.

 @willem2 beat me to it :D

Twisting any of the faces 180 degrees will have order 2, the smallest possible non-identity order.
Thank you. These are kind of uninteresting. Any interesting examples ... :)

If you have an element G of order n, you can easily find an element with an order of any divisor d of n
$$G^{n/d}$$
Is it easy to find? For example 10 divides the 1260 (order of
). How I am going to find an element of order 10 from this ... ??

jbunniii
Homework Helper
Gold Member
Is it easy to find? For example 10 divides the 1260 (order of
). How I am going to find an element of order 10 from this ... ??
Apply $(RU^2D^{-1}BD^{-1})$ 1260/10 = 126 times. It will be good exercise for your wrists. :-)

Apply $(RU^2D^{-1}BD^{-1})$ 1260/10 = 126 times. It will be good exercise for your wrists. :)
I am not going to do that :)

Apply $(RU^2D^{-1}BD^{-1})$ 1260/10 = 126 times. It will be good exercise for your wrists. :)
But how I'll convert that position to L R U D notation?

But how I'll convert that position to L R U D notation?

Thank you very much. Now I know.

Thank you very much. Now I know.
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

This was actually a serious reply.
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. :)