## Group theory II

For the discussion of math. Duh.

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tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Group theory II

Actually, using a slightly better magma program than Order(G), it turns out that it is actually the trivial group. Well that was frustrating. The Knuth-Bendix Completion algorithm was used. Speaking of which, could you run the below code to check what the next group is?
>G<a,b>:=Group<a, b| a^2, b^3, (a*b)^7, (a,b)^10, ((a,b)^4*b)^8>;
>R := RWSGroup(G:MaxRelations:=100000, TidyInt:=1000);
>Order(R);
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

DavCrav
Posts: 251
Joined: Tue Aug 12, 2008 3:04 pm UTC
Location: Oxford, UK

### Re: Group theory II

I ran the code, and R fails to be confluent.

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Group theory II

Oh, have you tried with larger values of G:MaxRelations and TidyInt? Those were adapted from the code that shows that <a,b|a2, b3, (ab)7, [a,b]10, ([a,b]4b)7> is the trivial group. It may take a bit more to figure out what this group is.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

DavCrav
Posts: 251
Joined: Tue Aug 12, 2008 3:04 pm UTC
Location: Oxford, UK

### Re: Group theory II

I put an extra zero on the end of MaxRelations but it seemed to take a long time and then my connection died and my SSH session closed. The trouble with things like this is, there's no reason to think that it will take less storage than there are atoms in the Universe...

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Group theory II

Ah, well, I'll have to find a faster way then. It seems unlikely that the word problem will be unsolvable for a two generator group with only five relations.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Group theory II

I have been reading this article about the Knuth Bendix algorithm, and it turns out that sometimes most of the time, for infinite groups, the confluent system is actually infinite. Although, it does turn out that, in many cases, the infinite system is regular, so it may be possible to describe it finitely, and in some cases, by a change of the presentation (without actually changing the group), the system can be made finite. The problem is verifying confluence in the infinite systems (after you've gotten the system in the first place).

Edit: I have made some progress on the groups. I now know a really simple presentation for the Janko group J1, <a,b|a2, b3, (ab)7, [a,b]10, ([a,b]2[a,b2])6>. I also found a presentation for the second Janko group J2, <a,b|a2, b3, (ab)7, [a,b]10, ([a,b]3[a,b2])5>.

The group <a,b|a2, b3, (ab)7, [a,b]10> must be quite interesting, since it has at least three simple groups as quotients, two of them sporadic groups. Someone at MO has checked, and PSL(2,41), J1, and J2 are the only quotients of order 500,000,000 or less.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

tomtom2357
Posts: 563
Joined: Tue Jul 27, 2010 8:48 am UTC

### Re: Group theory II

Just to give an update, we have checked up to 2,000,000,000, and there are no further simple groups so far. Also, there are no other simple quotients of the form PSL(2,p). We have calculated the abelianization of the kernel of the homomorphism from the group onto PSL(2,41), and it is elementary abelian of rank 42.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.