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Hello, My name is Jay. I am 21 years old with self taught knowledge in mathmatics and Group Theory. I've lurked this website for some time now, understand most of the topics of disscussion, and now pose a question to further my own understanding. So given all the information from this website:

Q: Given the Alg. RU'R'U'RUR'F'RUR'U'R'FR, How does the Alg. effect the cube?

I don't know if you will call this proof, but there are some restrictions to the cube.
1: There can never /just/ be two edges that are wrong
2:------------------------------corners--------------
3:There can be three edges that are wrong
4:-------------------corners-------------

Then I guess what I'm really trying to ask is based on this:
--------
||1 2 3 ||
||4 A 5 ||
||6 7 8 ||
---------------------------------------------
||9 10 11||17 18 19 ||25 26 27||33 34 35||
||12 B 13||20 C 21 ||28 D 29 ||36 E 37 ||
||14 15 16||22 23 24||30 31 32||38 39 40||
-----------------------------------------------
||41 42 43||
||44 F 45 ||
||46 47 48||
--------
Cycle Structure:
Where -> (A,B,C,D,E,F) = (U,L,F,R,B,D)

and R = (25,27,32,30) = D
(26,29,31,28) = D
( 3,38,43,19) = D
( 5,36,45,24) = D
Find L,U,D,F,B such that G(x) = (RU'R')(U'RU)[R'{F'(R(UR'U')R')F}R] = 0

Then I guess what I'm really trying to ask is based on this:
--------
||1 2 3 ||
||4 A 5 ||
||6 7 8 ||
---------------------------------------------
||9 10 11||17 18 19 ||25 26 27||33 34 35||
||12 B 13||20 C 21 ||28 D 29 ||36 E 37 ||
||14 15 16||22 23 24||30 31 32||38 39 40||
-----------------------------------------------
||41 42 43||
||44 F 45 ||
||46 47 48||
--------
Cycle Structure:
Where -> (A,B,C,D,E,F) = (U,L,F,R,B,D)

and R = (25,27,32,30) = D
(26,29,31,28) = D
( 3,38,43,19) = D
( 5,36,45,24) = D
Find L,U,D,F,B such that G(x) = (RU'R')(U'RU)[R'{F'(R(UR'U')R')F}R] = 0

That's Ok. I will be using this forum and thread really to answer all the questions I have until either I work them out on paper and put them here, or someone else beats me to it.

That's Ok. I will be using this forum and thread really to answer all the questions I have until either I work them out on paper and put them here, or someone else beats me to it.

I'm a little confused at what you're trying to get at. I'm guessing it's some sort of deeper understanding than "it just works because it works." Most people on this forum don't give any thought to algorithms past "what does it do?" and "can I do it fast?"

Matt was just writing the alg in commutator/conjugate notation. "," denotes a commutator and ":" denotes a conjugate. When expanded they are the same alg.

The reason why I posted this topic in the Puzzle Theory section is because I am trying to appeal to those who frequent the Puzzle Theory aspect of speedsolving. I too give thought to both "what it does" and "How fast can I do it", but you were also right in that I have other questions (puzzle theory questions) that need to be answered as well.

Matt was just writing the alg in commutator/conjugate notation. "," denotes a commutator and ":" denotes a conjugate. When expanded they are the same alg.

This is an acceptable answer to me. Actually, any answer is an acceptable answer, but please try to understand that I will most likely have more questions.

This is another great answer! Could you prove this answer using an "If... than..." statement where:

Given: (v,r,w,s)

V = (0,0,0,0,0,0,0,0) <-Orientation of the corners
R = (n_Sub_0,n_sub_1) <-Permutation of the corners
w = (0,0,0,0,0,0,0,0,0,0,0,0) <-Orientation of the edges
s = (I_Sub_S_Sub_12) <-Permutation of the edges

What's the purpose of renaming perfectly good things? And B, D and F appearing on both sides results in C=F=D=R and E=B=L which makes no sense whatsoever.

To treat the cube as a group, it's usually easiest to label every sticker with a number, and define the moves as permutations. One such definition of the permutations is here. From that, you can calculate the effect on a given piece as a composition of permutations. This gives an overall permutation, from which you can find cycle structure, etc.
There are a few threads about this, and I have a partial explanation on my website.

As Kirjava mentioned, to figure out the effect on a supercube, if you have an alg in a form without rotations, you can just count the total effect on each center by adding the moves on that side, mod 4.

This is also an excellent answer! The decompisition that I came up with is:
If G(x) = R U' R' U' R U R' F' R U R' U' R' F R
Than the decompisition of G(x) = (RU'R')(U'RU)[R'{F'(R(UR'U')R')F}R]

If this is true, how do I prove that this is SuperCube Safe?
I understand that the first part is a 4 cycle of those specific edges and corners, and that the second part is a 3 cycle of those specific edges and corners, but what about those two braketed off parts of the Alg. effect the orientation and permutation of those specific edges and corners as it relates to the rubiks cube as a Group G(x)?

The way I see it, as the corner pieces in positions (25,27,32,30) undergo the move R they change on the D face, which as you noticed is a redundant way of saying R face, to the positions on the cube marked by (26,29,31,28). Does this clairify?

As everyone else said, for the sequence to be supercube-safe, the amount of quarter turns in a defined direction per each face must be a multiple of 4. (because the centers are not exchangeable by face turns, on a 3x3).

Simply, R + R' + R + R' + R + R' + R' + R = 0*R
and U' + U' + U + U + U' = -1*U
and F' + F = 0*F

So one iteration of this sequence will rotate just the U center counterclockwise (while swapping UL+UB and ULB+RDF). This is not supercube-safe.

And I'm no good with groups, so I can't answer your G(x) question soz pl0x .