# Message #3975

From: Roice Nelson <roice3@gmail.com>

Subject: Re: [MC4D] Puzzle Group Centers and Superflips [3 Attachments]

Date: Fri, 19 Jan 2018 22:37:12 -0600

It appears I messed up the KQ superflip picture (it has some unintended

transparency). A nicer version without that problem is here:

https://photos.app.goo.gl/Z0PgeEZAJBx8wtS62

Best,

Roice

On Fri, Jan 19, 2018 at 10:09 PM, Roice Nelson roice3@gmail.com [4D_Cubing]

<4D_Cubing@yahoogroups.com> wrote:

> [Attachment(s) <#m_-3120951100068111505_TopText> from Roice Nelson

> included below]

>

> Hi all,

>

> I looked into the 3^4 group center

> <https://en.wikipedia.org/wiki/Center_(group_theory)> question using

> GAP. I wrote a short program to spit out the generators as a permutation

> group that GAP can read. I’ve attached files for both the 3^4 group and

> the Klein Quartic puzzle group, in case you want to investigate other

> properties yourself (I found this Rubik’s cube example

> <https://www.gap-system.org/Doc/Examples/rubik.html> useful). You can

> load these files with the GAP Read command. Once loaded, grab the size of

> the group in GAP with:

>

> Size( puzzle );

>

> Calculate and display the center of the group with:

>

> z := Centre( puzzle );

> GeneratorsOfGroup( z );

>

> For both puzzles, the group centers are like the original Rubik’s cube.

> They have only one non-trivial element that flips all 2-colored pieces. In

> the case of the 3^4, I was hoping the "superflip" move would also reorient

> the corners in place with double swaps. That seemed pretty but alas, it

> only reorients the 24 2C pieces.

>

> If you aren’t familiar with the center of a group, it is a subgroup that

> contains all elements that commute with every element of the group. So if

> you made a macro to execute the superflip S, then for any other sequence of

> moves X, SXS’ = X. (Since these superflips are order 2, S is also its own

> inverse, S = S’.) If a center is the entire group, the group is abelian

> and probably not an interesting permutation puzzle. You might say

> (roughly) that groups with small centers are less abelian.

>

> It is interesting to me that GAP can quickly find the center in a group

> with so many elements. Also worthy of note: it took just a few seconds to

> find it for the 3^4, but a few minutes for KQ. I don’t know why.

>

> *Conjecture in analogy to the 3^3*: The KQ and 3^4 superflips are as far

> away from pristine as the diameter of the state space. That is,

> it requires at least God’s number of moves to get to these positions. Also

> in analogy to the 3^3, I bet it will be much easier to verify the minimum

> number of moves to reach a superflip than to verify that this count is

> God’s number.

>

> Cheers,

> Roice

>

>

>

>