Primitive Cayley Graphs

What is a transitive group?

A group is transitive if any element can be mapped to any other element.

NOT THIS:

"<g> = G for some g"

What is a regular group?

A group is regular if only the identity fixes some element.

NOT THIS:

"Stab(g) = {1}"

What is a primitive permutation group?

A permutation group G acting on a set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X.

NOT THIS:

Theorem: (Density of primitive permutation groups)

For almost all positive integers n, the only primitive permutation groups on a set of size n are         and       .

A_n
A​n​​
S_n
S​n​​

"Every transitive group of prime degree is primitive."

Cayley Graph

\text{Cay}(G, S)
Cay(G,S)
V = \{g|g \in G\}
V={g∣g∈G}
E = \{(a, b) |as = b; a, b \in G, s \in S\}
E={(a,b)∣as=b;a,b∈G,s∈S}
\langle(01), (23), (45)\rangle
⟨(01),(23),(45)⟩
S \subseteq S^{-1}
S⊆S​−1​​

Arthur Cayley (1821-1895)

  • Sadleirian professorship at Cambridge in pure mathematics
  • Worked to advance women's education
  • Cayley's theorem
  • Cayley Hamilton theorem (every square matrix is root of its own characteristic polynomial)
  • Name associated with the following types of mathematical objects: theorem, algebra, determinant, diagram, construction, graph, number, sextic, table, algorithm, formula, metric, model, process, surface, transform

Why are they cool/useful?

  • They provide insight into group structure by encoding the symmetries of a set of actions into
  • Examples?
\text{Aut}(\text{Cay}(G, S))
Aut(Cay(G,S))

FACTS

  • All Cayley graphs are vertex-transitive
  • Not all vertex-transitive graphs are Cayley graphs
  •       is a Cayley graph if and only if there exists regular subgroup

 

  • There are 15,506 vertex-transitive graphs for                   , of which 15,394 are Cayley graphs

Conjecture: (McKay Praeger)

For almost all     , vertex transitive graphs are Cayley graphs  (with                    .

\Gamma
Γ
R \le \text{Aut}(\Gamma)
R≤Aut(Γ)
n
n
n = |G|
n=∣G∣
n = 24
n=24

Strategy

  1. Find those vertex transitive graphs whose automorphism group contains a regular subgroup (this is hard?)
R \le \text{Aut}(\Gamma)
R≤Aut(Γ)

But we just saw that Cayley graphs are super common!

 

Instead of finding Cayley graphs, maybe we should find non-Cayley graphs!

is regular

Non-Cayley Project

  • Non-Cayley Project: determine all positive integers      such that all vertex-transitive graphs on       vertices are Cayley graphs. (E.g.                                         )
n = p^i, i \in \{1, 2, 3\}
n=p​i​​,i∈{1,2,3}
n
n
n
n

Primitive Cayley Graph

                        is primitive if the only vertex partitions under                                  are trivial partitions.

\text{Cay}(G, S)
Cay(G,S)
\text{Aut}(\text{Cay}(G, S))
Aut(Cay(G,S))

"the complete graph         is a primitive Cayley graph"

K_n
K​n​​

Burnside's Discovery

"The only primitive Cayley graphs for cyclic groups of non-prime order are complete graphs."

If a primitive group contains a cyclic regular subgroup of non-prime order, the group is 2-transitive.

K_n
K​n​​

FACT: Only          has 2-transitive action on vertices of a connected graph under graph automorphism.

MORE FACTS

Theorem: (Cameron, Neumann, and Teague)

For almost all    , the only primitive permutation groups on a set of size     are         and

n
n
n
n
A_n
A​n​​
S_n
S​n​​

All finite primitive permutation groups containing regular cyclic subgroup (Gareth Jones).

helps find non-Cayley

All the finite primitive permutation groups H containing a regular abelian or dihedral subgroup (Cai Heng Li).

helps find Cayley

Theorem: (G = finite non-abelian simple group)

Let Cay(G, S) be a primitive Cayley graph for a finite non-abelian simple group G. Then either S is a union of conjugacy classes of G, or G =       for some prime                         .

A_{p^2 - 2}
A​p​2​​−2​​
p = 3 \text{mod} 4
p=3mod4

"Thus the primitive Cayley graphs for finite nonabelian simple groups are essentially well understood."

REFRINSEZ

Praeger, Cheryl E. "Regular permutation groups and Cayley Graphs." WSPC Proceedings, 12 March 2009

 

"Primitive Permutation Group." Wikipedia. Wikimedia Foundation, n.d. Web. 05 May 2015.

 

"Noncayley Graph." -- from Wolfram MathWorld. N.p., n.d. Web. 06 May 2015.

What is a primitive permutation Cayley graph?

Cay(G, S) is primitive if it preserves no nontrivial partition of X

A group is regular if the kernel of any element is trivial

These are the smallest transitive groups

2-Transitive means any pair of elements may be mapped to any other pair

All Cayley graphs are vertex transitive

A graph is a Cayley graph iff exists subgroup R of Aut regular on vertices