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.
Best Answer
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.