Sims conjecture

{{short description|Conjecture in group theory}}

In mathematics, the Sims conjecture is a result in group theory, originally proposed by Charles Sims.{{cite journal|last=Sims |first=Charles C. |authorlink=Charles Sims (mathematician) |title=Graphs and finite permutation groups |journal=Mathematische Zeitschrift |volume=95 |number=1 |year=1967 |pages=76–86 |doi=10.1007/BF01117534|s2cid=186227555 }} He conjectured that if $G$ is a primitive permutation group on a finite set $S$ and $G\_\backslash alpha$ denotes the stabilizer of the point $\backslash alpha$ in $S$, then there exists an integer-valued function $f$ such that $f(d)\; \backslash geq\; |G\_\backslash alpha|$ for $d$ the length of any orbit of $G\_\backslash alpha$ in the set $S\; \backslash setminus\; \backslash \{\backslash alpha\backslash \}$.

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.{{Cite arXiv|eprint = 2102.06670|last1 = Pyber|first1 = László|last2 = Tracey|first2 = Gareth|title = Some simplifications in the proof of the Sims conjecture|year = 2021| class=math.GR }}

{{math theorem|There exists a function $f:\; \backslash mathbb\{N\}\; \backslash to\; \backslash mathbb\{N\}$ such that whenever $G$ is a primitive permutation group and $h\; >\; 1$ is the length of a non-trivial orbit of a point stabilizer $H$ in $G$, then the order of $H$ is at most $f(h)$.}}

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.{{cite journal|last1=Cameron |first1=Peter J. |last2=Praeger |first2=Cheryl E. |last3=Saxl |first3=Jan |last4=Seitz |first4=Gary M. |authorlink1=Peter Cameron (mathematician) |authorlink2=Cheryl Praeger |author3-link=Jan Saxl|authorlink4=Gary Seitz |title=On the Sims conjecture and distance transitive graphs |journal=Bulletin of the London Mathematical Society |volume=15 |year=1983 |issue=5 |pages=499–506 |doi=10.1112/blms/15.5.499}}{{cite journal|last=Cameron |first=Peter J. |authorlink=Peter Cameron (mathematician) |title=There are only finitely many distance-transitive graphs of given valency greater than two |journal=Combinatorica |year=1982 |volume=2 |number=1 |pages=9–13 |doi=10.1007/BF02579277|s2cid=6483108 }}{{cite book|last=Isaacs |first=I. Martin |authorlink=Martin Isaacs |title=Finite Group Theory |publisher=American Mathematical Society |year=2011 |isbn=9780821843444 |oclc=935038216}}