Results in the Presentation of Finite Groups - MathOverflow

Results in the Presentation of Finite Groups

2012-03-07T21:31:27Z

I've been looking at combinatorial group theory, but all the results seem to be about infinite groups. Are there any important results about the presentations finite groups specifically (or are useful for finite groups?). About how the minimum number of relations implies something about the structure of the group?

I'd prefer results that are applicable to all finite groups or to all finite simple or all simple groups.

Answer by Mark Sapir for Results in the Presentation of Finite Groups

2012-03-07T22:01:52Z

Yes, of course, here are a few examples:

Finite Coxeter groups (see any book on Coxeter groups), they are given by their presentations.
Finite simple groups, see http://pages.uoregon.edu/kantor/PAPERS/GKKL2.pdf and references there.

Answer by Derek Holt for Results in the Presentation of Finite Groups

2012-03-07T22:23:27Z

One such result that springs to mind is that, if the finite group $G$ has a presentation with $r$ generators and $s$ relations, then the Schur Multiplier $M(G) = H_2(G)$ of $G$ can be generated by at most $s-r$ elements. So in particular $s \ge r$. (Of course you can prove that more directly - a finitely presented group with $s<r$ has infinite abelianization.)

We can deduce for example that a finite abelian group $G$ of rank $r$ requires at least $r(r+1)/2$ relations to present it, because $M(G)$ has rank $r(r-1)/2$. In this case the converse holds - the obvious presentation of $G$ has $r$ generators and $r(r+1)/2$ relations.

The converse does not hold in general. Swan constructed examples of finite solvable groups with trivial multiplier and arbitrarily large minimum $s-r$. But I believe it is still an open problem for finite $p$-groups. i.e. does every $d$-generator finite $p$-group have a presentation with $d$ generators and $d + {\rm rk}(M(G))$ relations?

Finite groups of defect 0 - i.e. $r=s$ have also been much studied. There are lots of 2-generator examples known, a few 3-generator examples and none requiring 4 or more generators.

Answer by Tom De Medts for Results in the Presentation of Finite Groups

2012-03-08T09:03:22Z

The result by Guralnick, Kantor, Kassabov and Lubotzky (which is mentioned already in Mark Sapir's answer) is one of the most striking results in finite group theory of the last decade, in my opinion. It shows that there is a uniform bound on the length of the presentation for (probably all) nonabelian finite simple groups. This is much stronger than what anyone before this result even conjectured, so it's absolutely amazing. More precisely, they show the following:

All nonabelian finite simple groups of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2$, have presentations with at most 80 relations and bit-length $O(\log n + \log q)$.

(By the way, the assumption "nonabelian" is essential, the theorem is false for cyclic groups of prime order!) Their results appeared in a series of 3 papers:

I would recommend you to have a look at the first of those 3 papers, it contains a wealth of information about presentations of finite (simple) groups, and it is written in a very readable and enjoyable style.