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 ${\displaystyle \left(G,\circ \right)}$, and any set of generators S. Define ${\displaystyle D_{S}}$ to be the graph diameter of the Cayley graph ${\displaystyle \Lambda =\left(G,S\right)}$. Then the diameter of ${\displaystyle \left(G,\circ \right)}$ is the largest value of ${\displaystyle D_{S}}$ 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 ${\displaystyle \lfloor s/2\rfloor }$.[1]

It is conjectured, for all non-abelian finite simple groups G, that[2]

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

Many partial results are known but the full conjecture remains open.[3]

References

1. Babai, László; Seress, Ákos (1992), "On the diameter of permutation groups", European Journal of Combinatorics, 13 (4): 231–243, arXiv:1109.3550, doi:10.1016/S0195-6698(05)80029-0, MR 1179520.
2. Babai & Seress (1992), Conj. 1.7. This conjecture is misquoted by Helfgott & Seress (2014), who omit the non-abelian qualifier.
3. Helfgott, Harald A.; Seress, Ákos (2014), "On the diameter of permutation groups", Annals of Mathematics, Second Series, 179 (2): 611–658, arXiv:1109.3550, doi:10.4007/annals.2014.179.2.4, MR 3152942.