Diameter (group theory)

In the area of abstract algebra known as group theory, the diameter of a finite group is a measure of its complexity.

Consider a finite group $$\left(G,\circ\right)$$, and any set of generators $S$. Define $$D_S$$ to be the graph diameter of the Cayley graph $$\Lambda=\left(G,S\right)$$. Then the diameter of $$\left(G,\circ\right)$$ is the largest value of $$D_S$$ taken over all generating sets $S$.

For instance, every finite cyclic group of order $s$, the Cayley graph for a generating set with one generator is an $s$-vertex cycle graph. The diameter of this graph, and of the group, is $$\lfloor s/2\rfloor$$.

It is conjectured, for all non-abelian finite simple groups $G$, that



\operatorname{diam}(G) \leqslant \left(\log|G|\right)^{\mathcal{O}(1)}. $$

Many partial results are known but the full conjecture remains open.