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Sorry folks, there was a slight error in my last message about the

maximum order of any element in the cube group. To understand why,

let us examine the transformation that ED described. I like to label

the cube this way:

A0 AE A1 AB AX AD A3 AC A2

B0 BA B3 C3 CA C2 D2 DA D1 E1 EA E0 BE BX BC CB CX CD DC DX DE ED EX EB B7 BF B4 C4 CF C5 D5 DF D6 E6 EF E7

F4 FC F5 FB FX FD F7 FE F6

(I won't bother to describe the features of this labeling, and I will

presume that the correspondence between this unfolded labeling and an

actual 3D cube is obvious.)

If I understand ED's directions this is the way the cube looks after

applying his transformation:

A3 AB A0 AC AX AE D1 DA D6

C3 CA E1 A1 AD F6 E6 EA E0 B0 BA B3 CB CX BF FB DX DE ED EX EB BE BX BC C2 EF E7 B7 CF B4 C4 DC C5 D5 DF D2

F7 FC F4 FE FX CD A2 FD F5

Note that the center faces BX, CX, DX and EX have all moved. My proof

that the maximum order of any element is 1260 assumes that these

center faces remain fixed. There is nothing wrong with that

assumption, it simply relects the intuition that picking a cube up and

putting it down again on a different face doesn't change the

configuration at all. If we decide that the orentation of the cube

DOES matter, then we get transformations like ED's here. The group

we are dealing with is also 24 times larger than before. And my proof

now shows that no element has an order greater than 2520 (twice as big).

Could it be that ED's transformation is not actually a maximal one?

Could there be one with higher order? Or can my proof be tightened up

some, so that even in the larger group the maximum order remains the

same?

To answer these questions (hopefully) I present a transformation

that I claim has order 2520:

D2 AB A0 DC AX AE D5 AD A1

C2 CD C5 F5 DA D1 E1 EA E0 B0 BA A2 CA CX CF FC DX DE ED EX EB BE BX AC C3 CB C4 F4 DF D6 E6 EF E7 B7 BF A3

B4 FD F6 BC FX FE B3 FB F7

(This transformation, like ED's, is not a member of the usual group,

but the larger one where the center faces are allowed to move.)

It is easy to check that this permutation has order 2520, the real

question is whether you can get to this configuration from a solved

cube. I haven't actually tried to do that, but I am fairly certain

that it is possible. If anyone would like to try it, and report to me

their success or failure, I would be very grateful.