[GAP Forum] Rubiks cube and Fp groups

[Rudolf Zlabinger]

Dear Mr. Hulpke,

thank you for your quick answer. I was not surprised to see, that there are
random algorithms in use, i would have used also for similar purposes. But
the idea, that the relators could contribute, was too attractive for me.

Semidirect product is correct in this case, where S(8) doesn't  p e r m u t
e  the components of cycle^8, as defined for wreath products in general, so
the only permutation on cycle^8 components would be the identity. My
question was not formulated correctly and therefore misleading. But in our
context there is no ambiguity at all, as I can see now.

As you can see, I had (erroneously?) some more general definition of "wreath
product" in mind, where the power of first factor is somewhat independent
from the cardinality of workingset of the second factor, and semidirect
product would be a special case. The action on the components of first
factor by second factor had to be defined separately in this case.

In the meantime i found, that "wreath product" has a more strict meaning at
least in the context of GAP and from there my question was already answered.

thank you and best wishes, Rudolf Zlabinger


----- Original Message ----- 
From: "Alexander Hulpke" <hulpke at mac.com>
To: "Rudolf Zlabinger" <Rudolf.Zlabinger at chello.at>
Cc: <forum at gap-system.org>
Sent: Saturday, April 01, 2006 6:36 PM
Subject: Re: [GAP Forum] Rubiks cube and Fp groups


Dear GAP Forum,

On Apr 1, 2006, at 6:49 AM, Rudolf Zlabinger wrote:
> I tested the presentation of a random element of the cubes
> permutations in a
> finitely presented group.
>

> The Preimage for
> the Free group gave a solution of length 120, the same performed
> with the Fp
> group, same permutation, resulted in a chain of 84 moves.

>
> The algorithm mentioned in the sample was using stabilizer chains.
> Is it the
> same for the Fp group I used for my test? Or is there help for a
> "better"
> algorithm caused by the relators? Or is it pure random behaviour?

The algorithm is the same. I uses random words, thus the different
length will be due to random behavior. If you construct the same
homomorphism anew and try you will see some length discrepancies.

>
> By the way, another question to the same sample. There are given "
> wreath
> products of a 3 cycle (2 cycle) with S(8)". Is it right to
> interprete them
> as wreath products of the cycles ^ 8 and S(8)?
I suppose you mean ``semidirect'' in the last line. Then yes.

Best wishes,

     Alexander Hulpke