Some More Discrete Mathematics
Alexander Lubotzky · Birkhäuser Basel eBooks · 1994
This chapter is devoted to several results on graphs or groups which are related to the topics discussed above. We begin with an application to finite simple groups: Every finite simple non-abelian group G has a set S of at most seven generators with respect to which every element of G can be written as a word of length O (log |G|) with elements from S ∪ S −1. This theorem is proved in Babai-KantorLubotzky [BKL] by finite group theoretic methods but with «unnatural» generators. It turns out that Property (T) and Selberg’s Theorem give similar results in special cases with «natural» generators. The use of these deep theorems seems at this point to be unavoidable.