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Robert Furber
Robert Furber
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This is a bit too long for a comment, so it's an answer. The Loops package for Gap, by Gabor Nagy and Petr Vojtechovsky contains implementations of all the nonassociative Moufang loops of order $\leq 64$ and order equal to $81$ or $243$. So I wrote a gap scriptgap script to calculate the association probabilities of triples of elements. I made it as simple-minded as possible to reduce the possibility of bugs, and because I'd never written anything in Gap before.

None of the association probabilities exceeded $\frac{43}{64}$, the association probability for the octonion loop, so the conjecture is correct for these particular Moufang loops (the script took about half an hour on my laptop).

Since the package also has some Bol loops, I checked them, and the left Bol loop of order 8 with the following Cayley table has association probability $\frac{13}{16} = \frac{52}{64} > \frac{43}{64}$:

  1 2 3 4 5 6 7 8
  ---------------
1|1 2 3 4 5 6 7 8
2|2 1 4 3 7 8 5 6
3|3 4 1 2 6 5 8 7
4|4 3 2 1 8 7 6 5
5|5 6 7 8 1 2 3 4
6|6 8 5 7 3 1 4 2
7|7 5 8 6 2 4 1 3
8|8 7 6 5 4 3 2 1

and many of the left Bol loops of order 16 also have association probabilities exceeding $\frac{43}{64}$. Therefore, if the conjecture is correct for Moufang loops, the proof must use an argument that fails for left Bol loops.

This is a bit too long for a comment, so it's an answer. The Loops package for Gap, by Gabor Nagy and Petr Vojtechovsky contains implementations of all the nonassociative Moufang loops of order $\leq 64$ and order equal to $81$ or $243$. So I wrote a gap script to calculate the association probabilities of triples of elements. I made it as simple-minded as possible to reduce the possibility of bugs, and because I'd never written anything in Gap before.

None of the association probabilities exceeded $\frac{43}{64}$, the association probability for the octonion loop, so the conjecture is correct for these particular Moufang loops (the script took about half an hour on my laptop).

Since the package also has some Bol loops, I checked them, and the left Bol loop of order 8 with the following Cayley table has association probability $\frac{13}{16} = \frac{52}{64} > \frac{43}{64}$:

  1 2 3 4 5 6 7 8
  ---------------
1|1 2 3 4 5 6 7 8
2|2 1 4 3 7 8 5 6
3|3 4 1 2 6 5 8 7
4|4 3 2 1 8 7 6 5
5|5 6 7 8 1 2 3 4
6|6 8 5 7 3 1 4 2
7|7 5 8 6 2 4 1 3
8|8 7 6 5 4 3 2 1

and many of the left Bol loops of order 16 also have association probabilities exceeding $\frac{43}{64}$. Therefore, if the conjecture is correct for Moufang loops, the proof must use an argument that fails for left Bol loops.

This is a bit too long for a comment, so it's an answer. The Loops package for Gap, by Gabor Nagy and Petr Vojtechovsky contains implementations of all the nonassociative Moufang loops of order $\leq 64$ and order equal to $81$ or $243$. So I wrote a gap script to calculate the association probabilities of triples of elements. I made it as simple-minded as possible to reduce the possibility of bugs, and because I'd never written anything in Gap before.

None of the association probabilities exceeded $\frac{43}{64}$, the association probability for the octonion loop, so the conjecture is correct for these particular Moufang loops (the script took about half an hour on my laptop).

Since the package also has some Bol loops, I checked them, and the left Bol loop of order 8 with the following Cayley table has association probability $\frac{13}{16} = \frac{52}{64} > \frac{43}{64}$:

  1 2 3 4 5 6 7 8
  ---------------
1|1 2 3 4 5 6 7 8
2|2 1 4 3 7 8 5 6
3|3 4 1 2 6 5 8 7
4|4 3 2 1 8 7 6 5
5|5 6 7 8 1 2 3 4
6|6 8 5 7 3 1 4 2
7|7 5 8 6 2 4 1 3
8|8 7 6 5 4 3 2 1

and many of the left Bol loops of order 16 also have association probabilities exceeding $\frac{43}{64}$. Therefore, if the conjecture is correct for Moufang loops, the proof must use an argument that fails for left Bol loops.

Source Link
Robert Furber
Robert Furber
  • 3.8k
  • 1
  • 23
  • 34

This is a bit too long for a comment, so it's an answer. The Loops package for Gap, by Gabor Nagy and Petr Vojtechovsky contains implementations of all the nonassociative Moufang loops of order $\leq 64$ and order equal to $81$ or $243$. So I wrote a gap script to calculate the association probabilities of triples of elements. I made it as simple-minded as possible to reduce the possibility of bugs, and because I'd never written anything in Gap before.

None of the association probabilities exceeded $\frac{43}{64}$, the association probability for the octonion loop, so the conjecture is correct for these particular Moufang loops (the script took about half an hour on my laptop).

Since the package also has some Bol loops, I checked them, and the left Bol loop of order 8 with the following Cayley table has association probability $\frac{13}{16} = \frac{52}{64} > \frac{43}{64}$:

  1 2 3 4 5 6 7 8
  ---------------
1|1 2 3 4 5 6 7 8
2|2 1 4 3 7 8 5 6
3|3 4 1 2 6 5 8 7
4|4 3 2 1 8 7 6 5
5|5 6 7 8 1 2 3 4
6|6 8 5 7 3 1 4 2
7|7 5 8 6 2 4 1 3
8|8 7 6 5 4 3 2 1

and many of the left Bol loops of order 16 also have association probabilities exceeding $\frac{43}{64}$. Therefore, if the conjecture is correct for Moufang loops, the proof must use an argument that fails for left Bol loops.