Diameter of a finite group

In group theory, the diameter of a group is a measure of a finite group's 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 maximal value of $D_S$ taken over all generating sets S.

It is conjectured that, for all finite simple groups G, that

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

References

• H. A. Helfgott and Á. Seress, "On the diameter of permutation groups", Annals of Mathematics, vol 179:2 (2014), pp 611–658; cf. arXiv:1109.3550