Things that are 'n-gen'?

[Kirjava]

Is there another term that can be used to describe a sequence of turns that involve N faces?

This is what we're missing.

I don't think it's /that/ vital though - there are plenty of other terms with multiple meanings that usually require minimal clarification.

 

[Cool Frog]

Comms are 2 gen. :O

 

[Jorghi]

Mathematically speaking, <RUR', D> is a two-generator group. If I write "<RUR', D>", then by definition I am considering RUR' and D to be generators of a group.

Historically, mathematicians have used {U, D, L, R, F, B} as a set of generators for what is generally called the "Rubik's Cube group." It is obvious from these 6 "moves" we can create any legal cube position by combining them in some way. RUR', for example, can be generated by RURRR. The "Rubik's Cube group" was defined as <U, D, L, R, F, B>. (The order of the generators is unimportant). Of course, there are many other ways the same group can be defined in terms of some generators.

Any group using only two of the above generators, such as <U, R>, then logically became to be called two-generator as only two of the six standard generators were being used.

A problem arises in the speedsolving community, because we often use moves that don't keep the centers in a fixed location. (In other words U may not always turn the layer with the same color center piece). We are now dealing with a mathematical group that is 24 times larger than the group that is usually termed "the Rubik's Cube group." Within this larger group, we have other moves that are "basic" enough to think of as "standard generators": M, x, r or Rw, etc. There is really no "standard" set of generators for this group. We could write <U, D, L, R, F, B, x, y, z> or <U, D, L, R, F, B, M, E, S> or <U, x, y>. Logically it tends to make sense to think of any move that can be performed as a single simple action like M, x, or R as a "standard generator" and then calling any combination of the original six generators and this extended set of standard generators as "2-gen". This does not change the fact, of course, that it is always possible (mathemtically speaking, anyway) to use as generators, elements that are combinations of the "standard" generators like RUR' or MUM'.

In any case, <R, U, L> is definitely three-gen. I suppose there exist two elements within this group that will generate the whole group, but simply writing it as <R, U, L> means we are treating this group as a group generated by 3 elements, and thus, we can not say it is two-generator even though only two axes are turned. Anybody who does not understand this simply doesn't understand the concept of n-generator groups.

This guy just demolished everyone lol. Thanks for the explanation. Time to create 2-5 gen f2l scrambles and find thing cool things.

 

[JonnyWhoopes]

This guy just demolished everyone and the OP lol. Thanks for the explanation. Time to create 2-5 gen f2l scrambles and find thing cool things.

You see, it's funny, 'cause nothing Stachu said disagreed with what Bruce said.

::EDIT:: Editing to clarify why I said what I did. Bolded in the quote. This is what the quote looked like before it was ninja-edited. You're fast Jorghi 0_0.

 

[StachuK1992]

Did you know:

<M,U,R,D,E,r> can solve everything?
U D' E' = y
R y2 r' y2 = x
R = R.

 

[Jorghi]

Explain please? Either I lack understanding or that's a mistake (probably the former).

I don't think R r= x he made a mistake maybe.

You see, it's funny, 'cause nothing Stachu said disagreed with what Bruce said.
::EDIT:: Editing to clarify why I said what I did. Bolded in the quote. This is what the quote looked like before it was ninja-edited. You're fast Jorghi 0_0.

Nope... I checked to see what Stachu said in case he was right. Thats why my post with edit was made before yours And why it doesn't say last edited on it either.

 

[cubernya]

Lol stachu...just remove the posts

By the way, isn't the E slice notation viewed from above?

 

[JonnyWhoopes]

By the way, isn't the E slice notation viewed from above?

Nope. Never understood why, it seems kinda non sequitur to me.

 

[vcuber13]

idk why m is the same as L either

 

[cubernya]

I thought the slices were as follows:

M - same as L
E - same as U
S - same as F

Correct me if I'm wrong

 

[StachuK1992]

http://www.speedsolving.com/wiki/index.php/Puzzle_Notation

 

[cubernya]

I could've sworn E was same as U

Edit: WCA regulations say E is same direction as D. Guess I'm going to have to get used to that

 

[Julian]

M = top to bottom
E = left to right
S = clockwise

 

[blah]

Didn't Lucas once prove that the entire cube group can be generated by a subgroup of order 2? Can't find the link though.

Edit: I refuse to use the term 2-gen in the middle of this noob confusion/ignorance.
Edit: Okay so he didn't prove it

 

[Sa967St]

Didn't Lucas once prove that the entire cube group was 2-gen? Can't find the link though.

http://www.speedsolving.com/forum/showthread.php?19751-The-entire-cube-is-2-gen!

 

[blah]

Not quite on-topic, but does anyone have a proof that the cube group is not cyclic? Does it involve permutation parities?

 

[qqwref]

Keep this in mind: in a mathematical context, generators are in terms of positions (piece permutations). A generator like <RUR', D> is valid and useful in a mathematical context because RUR' actually represents a single piece permutation. That generator contains every permutation that can be generated by those two permutations. In the cubing context, though, it's very imporatnt to think in terms of moves themselves, rather than cube positions, because otherwise we can't distinguish betwen R2URUR'U'R'U'R'UR' and F2UR'LF2RL'UF2. So in that spirit we should be thinking of generators in terms of individual moves. Allowing generators of multiple moves just dilutes the definition to the point where there is no meaningful distinction between 2-gen and not. From both a conceptual and a practical viewpoint, it makes more sense to be strict and allow N-gen to only contain individual moves.

tl;dr: We shouldn't consider things like <RUR', D> to be 2-gen.

 

[qqwref]

Not quite on-topic, but does anyone have a proof that the cube group is not cyclic? Does it involve permutation parities?

"The fundamental theorem of cyclic groups states that if G is a cyclic group of order n then every subgroup of G is cyclic."

EPLL (all 4!/2 cases) is a subgroup of G, and it is easy to see it is not cyclic (every element has order <12). QED.

 

[Kirjava]

in that spirit we should be thinking of generators in terms of individual moves. Allowing generators of multiple moves just dilutes the definition to the point where there is no meaningful distinction between 2-gen and not. From both a conceptual and a practical viewpoint, it makes more sense to be strict and allow N-gen to only contain individual moves.

Unless useful to do otherwise of course, in which case the meaning can be specified.

I like it when other people post what I'm thinking.

 

[JonnyWhoopes]

Keep this in mind: in a mathematical context, generators are in terms of positions (piece permutations). A generator like <RUR', D> is valid and useful in a mathematical context because RUR' actually represents a single piece permutation. That generator contains every permutation that can be generated by those two permutations. In the cubing context, though, it's very imporatnt to think in terms of moves themselves, rather than cube positions, because otherwise we can't distinguish betwen R2URUR'U'R'U'R'UR' and F2UR'LF2RL'UF2. So in that spirit we should be thinking of generators in terms of individual moves. Allowing generators of multiple moves just dilutes the definition to the point where there is no meaningful distinction between 2-gen and not. From both a conceptual and a practical viewpoint, it makes more sense to be strict and allow N-gen to only contain individual moves.

tl;dr: We shouldn't consider things like <RUR', D> to be 2-gen.

I am very okay with this.