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
>
>
>
>