Sims conjecture explained

In mathematics, the Sims conjecture is a result in group theory, originally proposed by Charles Sims.^[1] He conjectured that if

G is a primitive permutation group on a finite set S and G _{\alpha} denotes the stabilizer of the point \alpha in S , then there exists an integer-valued function f such that f(d) \geq |G _{\alpha|} for d the length of any orbit of G _{\alpha} in the set S \setminus \{\alpha\} .

The conjecture was proven by Peter Cameron, Cheryl Praeger, Jan Saxl, and Gary Seitz using the classification of finite simple groups, in particular the fact that only finitely many isomorphism types of sporadic groups 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 distance-transitive graphs having degree greater than 2.^{[3] [4] [5]}

Notes and References

1. Sims . Charles C. . Charles Sims (mathematician) . Graphs and finite permutation groups . . 95 . 1 . 1967 . 76–86 . 10.1007/BF01117534. 186227555 .
2. 2102.06670. Pyber. László. Tracey. Gareth. Some simplifications in the proof of the Sims conjecture. 2021. math.GR .
3. Cameron . Peter J. . Praeger . Cheryl E. . Saxl . Jan . Seitz . Gary M. . Peter Cameron (mathematician) . Cheryl Praeger . Jan Saxl. Gary Seitz . On the Sims conjecture and distance transitive graphs . . 15 . 1983 . 5 . 499–506 . 10.1112/blms/15.5.499.
4. Cameron . Peter J. . Peter Cameron (mathematician) . There are only finitely many distance-transitive graphs of given valency greater than two . . 1982 . 2 . 1 . 9–13 . 10.1007/BF02579277. 6483108 .
5. Book: Isaacs, I. Martin . Martin Isaacs

   . Martin Isaacs . Finite Group Theory . . 2011 . 9780821843444 . 935038216.