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 $(G,\circ )$, and any set of generators Template:Mvar. Define $D_S$ to be the graph diameter of the Cayley graph $\mathrm{\Lambda }=(G,S)$. Then the diameter of $(G,\circ )$ is the largest value of $D_S$ taken over all generating sets Template:Mvar.

For instance, every finite cyclic group of order Template:Mvar, the Cayley graph for a generating set with one generator is an Template:Mvar-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 Template:Mvar, that

$\mathrm{diam}(G)⩽{(\log |G|)}^{𝒪(1)}.$

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

References

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