 
Summary: Finite groups of uniform logarithmic diameter
Mikl´os Ab´ert
and L´aszl´o Babai
May 20, 2005
Abstract
We give an example of an infinite family of finite groups Gn such
that each Gn can be generated by 2 elements and the diameter of
every Cayley graph of Gn is O(log(Gn)). This answers a question of
A. Lubotzky.
Let G be a finite group and X a set of generators. Let Cay(G, X) denote
the undirected Cayley graph of G with respect to X, defined by having vertex
set G and g G being adjacent to gx±1
for x X.
We define diam(G, X) to be diameter of Cay(G, X), that is, the smallest
k such that every element of G can be expressed as a word of length at most
k in X (inversions permitted). The diameter of a Cayley graph is related to
its isoperimetric properties (cf. [1, 3, 7]).
Let us define the worst diameter of G,
diammax(G) = max {diam(G, X)  X G, X = G}
to be the maximum diameter over all sets of generators of G.
