In mathematics, the Sims conjecture is a result in

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, originally proposed by

Charles Sims Charles Sims may refer to: * Charles Sims (painter) (1873–1928), British painter * Charles Sims (mathematician) (1938–2017), American mathematician * Charles Sims (aviator) (1899–1929), British World War I flying ace * Charles Sims (America ...

. He conjectured that if

G

is a

primitive permutation group In mathematics, a permutation group ''G'' acting on a non-empty finite set ''X'' is called primitive if ''G'' acts transitively on ''X'' and the only partitions the ''G''-action preserves are the trivial partitions into either a single set or int ...

on a finite set

S

and

G_\alpha

denotes the stabilizer of the point

\alpha

S

, then there exists an integer-valued function

f

such that

f(d) \geq , G_\alpha,

for

d

the length of any

orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...

G_\alpha

in the set

S \setminus \

. The conjecture was proven by

Peter Cameron Peter Cameron is the name of: * Peter Cameron (entomologist) (1847–1912), English entomologist who specialised in Hymenoptera * Peter Cameron (minister) (born 1945), Scottish-born Church of Scotland minister convicted of heresy by the Presbyteria ...

, Cheryl Praeger,

Jan Saxl Jan Saxl (5 June 1948 – 2 May 2020) was a Czech-British mathematician, and a professor at the University of Cambridge. He was known for his work in finite group theory, particularly on consequences of the classification of finite simple groups ...

, and

Gary Seitz Gary Michael Seitz (born 1943) is an American mathematician, a Fellow of the American Mathematical Society and a College of Arts and Sciences Distinguished Professor Emeritus in Mathematics at the University of Oregon. He received his Ph.D. from t ...

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. 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 Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

greater than 2.

References

Algebraic graph theory Finite groups Permutation groups Theorems in graph theory Theorems in group theory {{graph-stub