Diameter (group theory)

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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 , and any set of generators S. Define to be the graph diameter of the Cayley graph . Then the diameter of is the largest value of 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 .[1]

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

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, MR 1179520, doi:10.1016/S0195-6698(05)80029-0 .
  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, MR 3152942, doi:10.4007/annals.2014.179.2.4 .


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