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Why does <R, U> preserve CP?

camicio

Member
Joined
Nov 18, 2011
Messages
4
What is the reason that doing only R and U moves preserves corner permutation? Well I guess "preserving" corner permutation isn't really the correct term; the more accurate question would be out of the 720 possible CP states with 6 corners, why are only 120 of them reachable using <R, U> moves? Why are the other 600 possible CP states not reachable with only <R, U> moves?
 
The symmetric group of degree 6, aka S_6, has non-inner automorphisms – isomorphisms to itself that aren't just a relabelling of the six objects being permuted.

It turns out that, viewed from the perspective of these outer automorphisms, the corner permutations induced by the R and U moves will fix one of the six objects (even though none of the six corners is fixed by both moves), so at most they can generate 5! = 120 permutations.

Further reading: https://en.wikipedia.org/wiki/Autom...oups#The_exceptional_outer_automorphism_of_S6
 
From Lemma 4 and Lemma 5 from my post, another way to look at it is that we cannot isolate a corner in the U or R face using only < R, U > moves. In other words, we cannot do 2-cycle or a 3-cycle of corners using only < R, U > moves. That will effectively reduce the number of possible corner permutations to 1/6 of what it would be otherwise.
 
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