# "Useful" permutation groups



## shadowslice e (Jun 19, 2015)

This may be a somewhat weird and vague question, but what how many groups are there that an 3x3 cube be reduced to which could be exploited or manipulated in a solve- in any discipline. For example an oh method may want to reduce the cube into the <r,u> group. 

Methods that already use this concept are those such as roux which essentially reduces the cube to the <u,m> group or altenatively zz which reduces to the <u,d,l,r> group In the first step.

I guess what I'm trying to ask is what states can the cube be reduced to that can meaningfully be acknowledged to help make to solving of the cube more ergonomic or efficient or even other applications I can't think of at the moment.

In addition, can these concepts be applied to big cubes (and possibly help direct solving methods) without first having to reduce to the <R,U,L,D,B,F> group (ie. the end product of the yau or reduction methods)?

Lastly, when considering these groups, would it be more useful to view corners and edges as essentially seperate systems or as simply as two sides of the same coin.

I don't expect a concrete answer but this is just really an idle query although should anything interesspting come out of this thread it could potentially become very beneficial to speedcubing in general- probably especially fmc.


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## Stefan (Jun 20, 2015)

Thistlethwaite and Kociemba use groups like <U,D,L2,R2,F2,B2>.


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## DizzypheasantZZ (Jun 20, 2015)

ZZ first step reduces to <R,U,L>.


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## spyr0th3dr4g0n (Jun 20, 2015)

All possible subgroups are useful if you have the correct alg set for it, some may just be more convenient for human solvers, others for computer solvers. As an example, if you scramble <f,r,u> ie wideturns, this is not too easy to solve compared to <F,R,U> for us, but I don't think it makes much difference for robotic solvers.
As for splitting them up or keeping them together, that's up to you, but note that inner slice moves such as <M,S,E> don't affect corner pieces, so a subset of that group can't solve corners on it own.


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## Lucas Garron (Jun 22, 2015)

<K, U> = <R U R', U>

Great for CLS.



Stefan said:


> Thistlethwaite and Kociemba use groups like <U,D,L2,R2,F2,B2>.



This is the most obvious one. Great for humans (BLD, various cuboids), great for computers.

See Thistlethwaite and Kociemba use groups like <U,D,L2,R2,F2,B2>.[/QUOTE]"]Human Thistlethwaite and Kociemba.


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