Sims conjecture

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Sims conjecture is a result in <a href="/facts/Group_theory/NfowcI5H">group theory</a>, originally proposed by <a href="/facts/Charles_Sims_(mathematician)/14FcNkhq">Charles Sims</a>. He <a href="/facts/Conjecture/oPWUb7SF">conjectured</a> that if 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a <a href="/facts/Primitive_permutation_group/Qq1nyEwq">primitive permutation group</a> on a finite set 
 
 
 
 S
 
 
 {\displaystyle S}
 
 and 
 
 
 
 
 G
 
 α
 
 
 
 
 {\displaystyle G_{\alpha }}
 
 denotes the <a href="/facts/Stabilizer_(group_theory)/Vh9VtUUw">stabilizer</a> of the point 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 in 
 
 
 
 S
 
 
 {\displaystyle S}
 
, then there exists an <a href="/facts/Integer/PUwwrawx">integer</a>-valued <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 such that 
 
 
 
 f
 (
 d
 )
 ≥
 
 |
 
 
 G
 
 α
 
 
 
 |
 
 
 
 {\displaystyle f(d)\geq |G_{\alpha }|}
 
 for 
 
 
 
 d
 
 
 {\displaystyle d}
 
 the length of any <a href="/facts/Orbit_(group_theory)/Vh9VtUUw">orbit</a> of 
 
 
 
 
 G
 
 α
 
 
 
 
 {\displaystyle G_{\alpha }}
 
 in the set 
 
 
 
 S
 ∖
 {
 α
 }
 
 
 {\displaystyle S\setminus \{\alpha \}}
 
. 
The conjecture was <a href="/facts/Mathematical_proof/7ECDrU80">proven</a> by <a href="/facts/Peter_Cameron_(mathematician)/sQnBHAhg">Peter Cameron</a>, <a href="/facts/Cheryl_Praeger/ssTafwxb">Cheryl Praeger</a>, <a href="/facts/Jan_Saxl/ydtFWjkA">Jan Saxl</a>, and <a href="/facts/Gary_Seitz/lnUkG1xo">Gary Seitz</a> using the <a href="/facts/Classification_of_finite_simple_groups/nMwLJf1K">classification of finite simple groups</a>, in particular the fact that only finitely many <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> types of <a href="/facts/Sporadic_group/hjNHTKVv">sporadic groups</a> exist.
The theorem reads precisely as follows.

Theorem—There exists a function 
 
 
 
 f
 :
 
 N
 
 →
 
 N
 
 
 
 {\displaystyle f:\mathbb {N} \to \mathbb {N} }
 
 such that whenever 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a primitive permutation group and 
 
 
 
 h
 >
 1
 
 
 {\displaystyle h>1}
 
 is the length of a non-trivial orbit of a point stabilizer 
 
 
 
 H
 
 
 {\displaystyle H}
 
 in 
 
 
 
 G
 
 
 {\displaystyle G}
 
, then the <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a> of 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is at most 
 
 
 
 f
 (
 h
 )
 
 
 {\displaystyle 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 <a href="/facts/Connectivity_(graph_theory)/FSNjsYhz">connected</a> <a href="/facts/Distance-transitive_graph/HgH4heaB">distance-transitive</a> <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graphs</a> having <a href="/facts/Degree_(graph_theory)/LnFFpI8v">degree</a> greater than 2.

Sims conjecture open-in-new

Sims conjecture