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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 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
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