# Sims conjecture

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Sims_conjecture
> Markdown URL: https://mediated.wiki/source/Sims_conjecture.md
> Source: https://en.wikipedia.org/wiki/Sims_conjecture
> Source revision: 1240488300
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [mathematics](/source/Mathematics), the **Sims conjecture** is a result in [group theory](/source/Group_theory), originally proposed by [Charles Sims](/source/Charles_Sims_(mathematician)).[1] He [conjectured](/source/Conjecture) that if G is a [primitive permutation group](/source/Primitive_permutation_group) on a finite set S and G_\alpha denotes the [stabilizer](/source/Stabilizer_(group_theory)) of the point \alpha in S, then there exists an [integer](/source/Integer)-valued [function](/source/Function_(mathematics)) f such that f(d) \geq |G_\alpha| for d the length of any [orbit](/source/Orbit_(group_theory)) of G_\alpha in the set S \setminus \{\alpha\}.

The conjecture was [proven](/source/Mathematical_proof) by [Peter Cameron](/source/Peter_Cameron_(mathematician)), [Cheryl Praeger](/source/Cheryl_Praeger), [Jan Saxl](/source/Jan_Saxl), and [Gary Seitz](/source/Gary_Seitz) using the [classification of finite simple groups](/source/Classification_of_finite_simple_groups), in particular the fact that only finitely many [isomorphism](/source/Isomorphism) types of [sporadic groups](/source/Sporadic_group) exist.

The theorem reads precisely as follows.[2]

Thus, in a primitive permutation group with "large" stabilizers, these stabilizers cannot have any small orbit. A consequence of their proof is that there exist only finitely many [connected](/source/Connectivity_(graph_theory)) [distance-transitive](/source/Distance-transitive_graph) [graphs](/source/Graph_(discrete_mathematics)) having [degree](/source/Degree_(graph_theory)) greater than 2.[3][4][5]

## References

1. Sims, Charles C. (1967). "Graphs and finite permutation groups". *[Mathematische Zeitschrift](/source/Mathematische_Zeitschrift)*. **95** (1): 76–86. [doi:10.1007/BF01117534](https://doi.org/10.1007/BF01117534). [S2CID 186227555](https://api.semanticscholar.org/CorpusID:186227555)

1. Pyber, László & Tracey, Gareth (2021). "Some simplifications in the proof of the Sims conjecture". [arXiv:2102.06670](https://arxiv.org/abs/2102.06670)

1. Cameron, Peter J.; Praeger, Cheryl E.; Saxl, Jan; Seitz, Gary M. (1983). "On the Sims conjecture and distance transitive graphs". *[Bulletin of the London Mathematical Society](/source/London_Mathematical_Society)*. **15** (5): 499–506. [doi:10.1112/blms/15.5.499](https://doi.org/10.1112/blms/15.5.499)

1. Cameron, Peter J. (1982). "There are only finitely many distance-transitive graphs of given valency greater than two". *[Combinatorica](/source/Combinatorica)*. **2** (1): 9–13. [doi:10.1007/BF02579277](https://doi.org/10.1007/BF02579277). [S2CID 6483108](https://api.semanticscholar.org/CorpusID:6483108)

1. Isaacs, I. Martin (2011). *Finite Group Theory*. [American Mathematical Society](/source/American_Mathematical_Society). ISBN 9780821843444. [OCLC 935038216](https://www.worldcat.org/oclc/935038216)

---
Adapted from the Wikipedia article [Sims conjecture](https://en.wikipedia.org/wiki/Sims_conjecture) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Sims_conjecture?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.