TITLE: Results in the Presentation of Finite Groups
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: 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:

Presentation of finite simple groups: a quantitative approach
Presentation of finite simple groups: cohomological and profinite approaches
Presentation of finite simple groups: a computational approach

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.