Diameter for a generating set of a group

Definition

In the group generating set sense

Suppose $G$ is a group and $S$ is a generating set for $G$. The diameter for $S$ in $G$ is defined in the following ways:

1. It is the diameter of the Cayley graph for $G$ with generating set $S$.
2. It is the maximum, over all elements $g\in G$, of the length of the smallest word that would give $g$ with the letters taken from the generating set $S$ and their inverses.

Facts

Note that there is a trade-off between the size of a generating set and the corresponding diameter. In general, the larger we make the generating set, the smaller the diameter. In particular, if the generating set is taken as the whole group, the diameter is 1.

There is a bound that says that the diameter for a generating set is roughly at least the ratio of the logarithm of the order of the group to the size of the generating set.

Generating sets that manage to make this trade-off well by coming close to the bound (i.e., by having a small size and a small diameter) are considered good. The corresponding Cayley graphs fit into the theory of expander graphs.