Sims conjecture

In mathematics, the Sims conjecture is a result in group theory, originally proposed by Charles Sims.[1] He conjectured that if ${\displaystyle G}$ is a primitive permutation group on a finite set ${\displaystyle S}$ and ${\displaystyle G_{\alpha }}$ denotes the stabilizer of the point ${\displaystyle \alpha }$ in ${\displaystyle S}$, then there exists an integer-valued function ${\displaystyle f}$ such that ${\displaystyle f(d)\geq |G_{\alpha }|}$ for ${\displaystyle d}$ the length of any orbit of ${\displaystyle G_{\alpha }}$ in the set ${\displaystyle S\setminus \{\alpha \}}$.

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. A consequence of their proof is that there exist only finitely many connected distance-transitive graphs having degree greater than 2.[2][3][4]

References

1. Sims, Charles C. (1967). "Graphs and finite permutation groups". Mathematische Zeitschrift. 95 (1): 76–86. doi:10.1007/BF01117534.
2. 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. 15: 499–506. doi:10.1112/blms/15.5.499.
3. Cameron, Peter J. (1982). "There are only finitely many distance-transitive graphs of given valency greater than two". Combinatorica. 2 (1): 9–13. doi:10.1007/BF02579277.
4. Isaacs, I. Martin (2011). Finite Group Theory. American Mathematical Society. ISBN 9780821843444. OCLC 935038216.