automorphism groups of cayley graphsmorris/banff-symmetries/talks/dobson-banfftalk.pdf · ted...

Automorphism Groups of Cayley Graphs

Ted Dobson

Department of Mathematics & StatisticsMississippi State [email protected]

http://www2.msstate.edu/∼dobson/

Symmetries of Graphs and Networks

November 23–28, 2008

Ted Dobson Mississippi State University

Basic Definitions

DefinitionLet G be a group and S ⊂ G . We define the Cayley digraph of G withconnection set S to be the digraph Γ = Γ(G ,S) defined by V (Γ) = G andE (Γ) = {(g , gs) : s ∈ S , g ∈ G}. If S = S−1, then Γ is a Cayley graph ofG with connection set S. Note that GL = {x → gx : g ∈ G} ≤ Aut(Γ).

A circulant graph of order n is simply a Cayley graph of Zn.

Basic Definitions

DefinitionLet G be a group and S ⊂ G . We define the Cayley digraph of G withconnection set S to be the digraph Γ = Γ(G ,S) defined by V (Γ) = G andE (Γ) = {(g , gs) : s ∈ S , g ∈ G}. If S = S−1, then Γ is a Cayley graph ofG with connection set S. Note that GL = {x → gx : g ∈ G} ≤ Aut(Γ).

A circulant graph of order n is simply a Cayley graph of Zn.

Automorphisms of prime order circulants

Observe that every vertex-transitive graph Γ of prime order is isomorphicto a circulant graph of prime order as Aut(Γ) contains a subgroup of orderp, which, after a relabeling, we may assume is〈(0, 1, . . . , p − 1)〉 = 〈x → x + 1〉 = (Zp)L. So circulant!

Theorem (Burnside, 1901)

Let G ≤ Sp, p a prime, contain (Zp)L. ThenG ≤ AGL(1, p) = {x → ax + b : a ∈ Z∗p, b ∈ Zp} or G is doubly-transitive.

Note that if G ≤ Aut(Γ) is doubly-transitive, then Aut(Γ) = SX and Γ iscomplete or has no edges.

This gives an algorithm for determining the full automorphism group of acirculant graph Γ = Γ(Zp,S). Note that x → x + b is always contained inAut(Γ), so we need only check which a ∈ Z∗p satisfya · S = {as : s ∈ S} = S (we observe that AGL(1, p) is itselfdoubly-transitive, so if all such x → ax are in Aut(Γ), then Aut(Γ) = Sp).

B. Alspach (1973) first observed this, and went on to explicitly enumeratethe vertex-transitive graphs of prime order.Burnside’s Theorem is equivalent to

TheoremLet G ≤ Sp, p a prime, be transitive. Then either G contains a normalSylow p-subgroup or G is doubly-transitive.

This gives an algorithm for determining the full automorphism group of acirculant graph Γ = Γ(Zp,S). Note that x → x + b is always contained inAut(Γ), so we need only check which a ∈ Z∗p satisfya · S = {as : s ∈ S} = S (we observe that AGL(1, p) is itselfdoubly-transitive, so if all such x → ax are in Aut(Γ), then Aut(Γ) = Sp).B. Alspach (1973) first observed this, and went on to explicitly enumeratethe vertex-transitive graphs of prime order.

Burnside’s Theorem is equivalent to

This gives an algorithm for determining the full automorphism group of acirculant graph Γ = Γ(Zp,S). Note that x → x + b is always contained inAut(Γ), so we need only check which a ∈ Z∗p satisfya · S = {as : s ∈ S} = S (we observe that AGL(1, p) is itselfdoubly-transitive, so if all such x → ax are in Aut(Γ), then Aut(Γ) = Sp).B. Alspach (1973) first observed this, and went on to explicitly enumeratethe vertex-transitive graphs of prime order.Burnside’s Theorem is equivalent to

Can Burnside’s Theorem be generalized?

Theorem (D., D. Witte, 2002)

There are exactly 2p − 1 transitive p-subgroups P of Sp2 up toconjugation, and all but three have the property that if G ≤ Sp2 withSylow p-subgroup P, then either P/G or G is doubly-transitive.

Theorem (D., 2005)

Let P be a transitive p-subgroup of Spk , p an odd prime, k ≥ 1, such thatevery minimal transitive subgroup of P is cyclic. If G ≤ Spk with Sylowp-subgroup P, then either P/G or G is doubly-transitive.

It is probable that many more generalizations are true, but it also seemslikely that “most classes” of automorphism groups of graphs cannot beobtained in this way.

Theorem (D., 2005)

Burnside’s Theorem can be restated as follows:

TheoremLet G be a transitive group of prime degree. Then G contains a transitivenormal subgroup which is either abelian or a nonabelian simple group.

The following result of Gareth Jones can also be viewed as an extension ofBurnside’s Theorem.

Theorem (Jones, 1979)

Let G ≤ Sp2 be transitive with Sylow p-subgroup isomorphic to Zp × Zp.Then G contains a normal subgroup H such that H = H1 × H2 where Hi

is a nonabelian simple group or Hi = Zp.

Theorem (D., 2008)

Let G ≤ Spk be transitive with an abelian Sylow p-subgroup P. Let t bethe number of elementary divisors of P. If p > 2t−1, then G contains anormal subgroup permutation isomorphic to a direct product of cyclicgroups and doubly-transitive nonabelian simple groups with the canonicalaction, with the number of factors in the direct product equal to thenumber of elementary divisors of P.

ProblemLet G ≤ Spk have an abelian Sylow p-subgroup P. Assume G admits anontrivial complete block system B such that StabG (B)|B is primitive andnot permutation isomorphic to a subgroup of AGL(r , p) for any r ≥ 1, butfixG (B)|B is imprimitive, B ∈ B. Then G contains a normal subgroup thatadmits a nontrivial complete block system C such that C ≺ B.

Theorem (D., 2008)

Let G ≤ Spk be transitive with an abelian Sylow p-subgroup P. Let t bethe number of elementary divisors of P. If p > 2t−1, then G contains anormal subgroup permutation isomorphic to a direct product of cyclicgroups and doubly-transitive nonabelian simple groups with the canonicalaction, with the number of factors in the direct product equal to thenumber of elementary divisors of P.

ProblemLet G ≤ Spk have an abelian Sylow p-subgroup P. Assume G admits anontrivial complete block system B such that StabG (B)|B is primitive andnot permutation isomorphic to a subgroup of AGL(r , p) for any r ≥ 1, butfixG (B)|B is imprimitive, B ∈ B. Then G contains a normal subgroup thatadmits a nontrivial complete block system C such that C ≺ B.

Using these results and other techniques, the full automorphism group of aCayley (di)graph of the following groups are known:

I Zpk , k ≥ 1. First proof using method of Schur by Klin and Poschel,in the early ’80’s. Another proof using result above. None of theproofs published...

I Zp × Zp, Dobson and D. Witte, 2002.

I Zp × Zp2 , Dobson, manuscript

I Z3p, Dobson and I. Kovacs, manuscript

I Zpk , k ≥ 1.

First proof using method of Schur by Klin and Poschel,in the early ’80’s. Another proof using result above. None of theproofs published...

I Zpk , k ≥ 1. First proof using method of Schur by Klin and Poschel,in the early ’80’s.

Another proof using result above. None of theproofs published...

GRR’s and DRR’s

DefinitionA Cayley digraph Γ of G is a digraphical regular representation (DRR) ofG if Aut(Γ) = GL.

A similar definition gives a graphical regular representation (GRR) of GIn 1978, C. Godsil finished off the problem of determining which groupshave DRR’s or GRR’s.

TheoremAll finite groups admit DRR’s with five exceptions of order at most 16.

TheoremAll finite groups admit GRR’s except abelian groups of exponent at least3, generalized dicyclic groups, and 13 other groups of order at most 32.

GRR’s and DRR’s

A similar definition gives a graphical regular representation (GRR) of G

In 1978, C. Godsil finished off the problem of determining which groupshave DRR’s or GRR’s.

GRR’s and DRR’s

The Conjecture

Motivated by work on determining which groups are DRR’s and GRR’s,several people became convinced that the following conjecture is true:

Conjecture

Let G be a finite group of order g which is neither abelian of exponent atleast 3 nor generalized dicyclic. Then

limg→∞

# of GRR′s of G

# of Cayley graphs of G= 1.

The conjecture is due to Imrich, Lovasz, Babai, and Godsil in around 1982.

There is also the similar conjecture that “almost all” Cayley digraphs of Gare DRR’s of G .

The Conjecture

Conjecture

limg→∞

# of GRR′s of G

The Conjecture

Conjecture

limg→∞

# of GRR′s of G

The Conjecture

Conjecture

limg→∞

# of GRR′s of G

The Conjecture

Conjecture

limg→∞

# of GRR′s of G

Known Results

The first results were obtained by Godsil in 1981:

TheoremLet G be a group of prime-power order with no homomorphism ontoZp o Zp (a full Sylow p-subgroup of Sp2). Then almost all Cayley digraphsof G are DRR’s of G.

TheoremLet G be a non-abelian group of prime-power order with nohomomorphism onto Zp o Zp (a full Sylow p-subgroup of Sp2). Thenalmost all Cayley graphs of G are GRR’s of G .

Known Results

These results were improved by Babai and Godsil in 1982:

TheoremLet G be a nilpotent group of odd order g. Let Γ be a random Cayleydigraph or Cayley graph of G . In the undirected case, assume additionallythat G is not abelian. Then the probability that Aut(Γ) 6= G is less than(0.91 + o(1))

√g .

TheoremLet G be an abelian group of order g ≡ −1 (mod 4). Then, for almost allCayley graphs Γ of G , |Aut(Γ)| = 2|G |.

√g .

Corollary

Let k be a positive integer, and p a prime such that p > 2k−1. LetG ≤ Spk be the automorphism group of a Cayley digraph of some abeliangroup P. Then one of the following is true:

1. G contains an automorphism α of P of order p, or

2. G has a normal Sylow p-subgroup which is isomorphic to P andabelian, or

3. G contains a transitive subgroup isomorphic to Sp × A, whereA ≤ Spk−1 has an abelian Sylow p-subgroup.

Automorphism groups of graphs

The following result extends Babai and Godsil’s result for graphs, providedthat the order of the group is an odd prime power.

TheoremLet G be an abelian group of odd prime-power order. Then almost everyCayley graph of G has automorphism group of order 2 · |G |.

Automorphism groups of graphs

The following result extends Babai and Godsil’s result for graphs, providedthat the order of the group is an odd prime power.

TheoremLet G be an abelian group of odd prime-power order. Then almost everyCayley graph of G has automorphism group of order 2 · |G |.

Normal Cayley Graphs

Ming-Yao Xu introduced the notion of a “normal Cayley graph” in 1998:

DefinitionA Cayley (di)graph Γ of G is a normal Cayley (di)graph of G if GL/Aut(Γ).

Xu also proposed a weaker conjecture to the one of Imrich, Lovasz, Babai,and Godsil:

Conjecture

Almost every Cayley (di)graph is a normal Cayley digraph.

Conjecture

We prove something more interesting, but not more surprising.

TheoremLet G be an abelian group of prime-power order pk . Then almost everyCayley digraph of G that is not a DRR is a normal Cayley digraph of G . Inparticular,

limp→∞

|NorCayDi(G )− DRR(G )||CayDi(G )− DRR(G )|

TheoremLet G be an abelian group of prime-power order pk . Then almost everyCayley graph of G that does not have automorphism group of order 2 · |G |is a normal Cayley graph of G .

limp→∞

Some Problems and ConjecturesWe would like to make the following (not terribly surprising) conjecture:

Conjecture

Almost every Cayley (di)graph whose automorphism group is not as smallas possible is a normal Cayley (di)graph.

It is difficult to determine the automorphism group of a (di)graph, so themain way to obtain examples of vertex-transitive graphs is to constructthem. An obvious construction is that of a Cayley (di)graph, and theconjecture of Imrich, Lovasz, Babai, and Godsil says that when performingthis construction, additional automorphism are almost never obtained. Theobvious way of constructing a Cayley (di)graph of G that does not haveautomorphism group as small as possible is to choose an automorphism αof G and make the connection set a union of orbits of α. The aboveconjecture in some sense says that this construction almost never yieldsadditional automorphisms other than the ones given by the construction.

Conjecture

It is difficult to determine the automorphism group of a (di)graph, so themain way to obtain examples of vertex-transitive graphs is to constructthem.

An obvious construction is that of a Cayley (di)graph, and theconjecture of Imrich, Lovasz, Babai, and Godsil says that when performingthis construction, additional automorphism are almost never obtained. Theobvious way of constructing a Cayley (di)graph of G that does not haveautomorphism group as small as possible is to choose an automorphism αof G and make the connection set a union of orbits of α. The aboveconjecture in some sense says that this construction almost never yieldsadditional automorphisms other than the ones given by the construction.

Conjecture

It is difficult to determine the automorphism group of a (di)graph, so themain way to obtain examples of vertex-transitive graphs is to constructthem. An obvious construction is that of a Cayley (di)graph, and theconjecture of Imrich, Lovasz, Babai, and Godsil says that when performingthis construction, additional automorphism are almost never obtained.

Theobvious way of constructing a Cayley (di)graph of G that does not haveautomorphism group as small as possible is to choose an automorphism αof G and make the connection set a union of orbits of α. The aboveconjecture in some sense says that this construction almost never yieldsadditional automorphisms other than the ones given by the construction.

Conjecture

It is difficult to determine the automorphism group of a (di)graph, so themain way to obtain examples of vertex-transitive graphs is to constructthem. An obvious construction is that of a Cayley (di)graph, and theconjecture of Imrich, Lovasz, Babai, and Godsil says that when performingthis construction, additional automorphism are almost never obtained. Theobvious way of constructing a Cayley (di)graph of G that does not haveautomorphism group as small as possible is to choose an automorphism αof G and make the connection set a union of orbits of α.

The aboveconjecture in some sense says that this construction almost never yieldsadditional automorphisms other than the ones given by the construction.

Conjecture

There are two additional families of (di)graphs that can be considered,namely “semiwreath products” and “deleted wreath products”

- suchgraphs are not normal Cayley graphs provided p 6= 2.

DefinitionA Cayley graph Γ of an abelian group G is a semiwreath product if thereexist subgroups H ≤ K < G such that S − K is a union of cosets of H.

Note that if H = K , a semiwreath product is in fact a wreath product.

DefinitionA Cayley graph Γ of an abelian group G is a deleted wreath product ifΓ = (Γ1 o Km)−mΓ1, where Γ1 is a Cayley graph of an abelian group oforder |G |/m, and mΓ1 is m vertex-disjoint copies of Γ1.

There are two additional families of (di)graphs that can be considered,namely “semiwreath products” and “deleted wreath products” - suchgraphs are not normal Cayley graphs provided p 6= 2.

Conjecture

Let Γ be a Cayley (di)graph of an abelian group G . Then one of thefollowing is true:

I Γ is a normal Cayley graph of G ,

I Γ is a semiwreath product, or

I the automorphism group of Γ is same as the automorphism group of adeleted wreath product.

The preceding conjecture is known to be true if G is cyclic.

The preceding conjecture is known to be false for at least some nonabeliangroups G .

Conjecture

ProblemFor an abelian group G , does there exist a natural collection F of familiesof Cayley (di)graphs of G and a partial order � on F such that everyCayley (di)graph of G is contained in some element of F and if F1 � F2

and there is no F3 such that F1 � F3 � F2, then almost every Cayley(di)graph of G that is not in F1 is in F2?

Automorphism Groups of Circulant Digraphs

The following result is a group-theoretic translation of several papers onSchur rings by Leung and Man (1996 and 1998), and Evdomikov andPonomarenko (2002), and appears in a paper by Li (2005).

TheoremLet G ≤ Sn contain a regular cyclic subgroup 〈ρ〉. Then one of thefollowing statements holds:

1. G (2) = G1 × G2 × · · · × Gr , where r ≥ 1, each Gi∼= Sni , or Gi

contains a normal regular cyclic group of order ni , such thatgcd(ni , nj) = 1 and n = n1n2 · · · nr , or

2. if G ≤ Aut(Γ) for a circulant digraph Γ, then Γ is a semiwreathproduct.

The same result was obtained by Dobson and J. Morris for square-freeintegers n in 2005.

Ponomarenko also gave a polynomial time algorithm (nO(1)) to determinethe full automorphism group of a circulant digraph in 2006,

while Dobsonand Morris have found an algorithm to determine the full automorphismgroup of a circulant digraph of square-free order in time O(n3 log n).

This may seem to end the story, but it does not. There are manyinteresting classes of circulant digraphs for which more precise informationcan be obtained, as there are usually many different possibleautomorphism groups of semiwreath circulants. We give several examples,the first due to I. Kovacs and independently to C. H. Li (2005):

TheoremLet Γ be a connected arc transitive circulant of order n which is not acomplete graph. Then either (1) Γ is a normal circulant, or (2) there existsan arc transitive circulant Γ1 of order m such that n = mb with b,m > 1and Γ = Γ1 o Kb, or Γ1 o Kb − bΓ1 with (b,m) = 1.

Ponomarenko also gave a polynomial time algorithm (nO(1)) to determinethe full automorphism group of a circulant digraph in 2006, while Dobsonand Morris have found an algorithm to determine the full automorphismgroup of a circulant digraph of square-free order in time O(n3 log n).

This may seem to end the story, but it does not.

There are manyinteresting classes of circulant digraphs for which more precise informationcan be obtained, as there are usually many different possibleautomorphism groups of semiwreath circulants. We give several examples,the first due to I. Kovacs and independently to C. H. Li (2005):

This may seem to end the story, but it does not. There are manyinteresting classes of circulant digraphs for which more precise informationcan be obtained, as there are usually many different possibleautomorphism groups of semiwreath circulants.

We give several examples,the first due to I. Kovacs and independently to C. H. Li (2005):

This may seem to end the story, but it does not. There are manyinteresting classes of circulant digraphs for which more precise informationcan be obtained, as there are usually many different possibleautomorphism groups of semiwreath circulants. We give several examples,

the first due to I. Kovacs and independently to C. H. Li (2005):

This may seem to end the story, but it does not. There are manyinteresting classes of circulant digraphs for which more precise informationcan be obtained, as there are usually many different possibleautomorphism groups of semiwreath circulants. We give several examples,the first due to I. Kovacs and independently to C. H. Li (2005):

The following results are in a manuscript by J.Araujo, Dobson, J.Konieczny, and J. Morris.

TheoremLet D be a circulant digraph of order n such that D is regular of degree atmost 2p − 2, where p is the smallest prime divisor of n. Then one of thefollowing is true:

1. D is a connected normal circulant digraph of Zn,

2. D has connection set (u + H)− {h}, whereH = {0, n/p, 2n/p, . . . , (p − 1)n/p}, u ∈ Z∗n, and h ∈ u + H suchthat h ≡ 0 (mod p), p2 does not divide n, and Aut(D) = Zn/p × Sp

(note that it is possible in this case that n/p = 1),

3. D has no edges or every edge is a loop and Aut(D) = Sn, or

4. D is disconnected, D = Km o D ′, where D ′ is a connected circulantdigraph of order k, mk = n (and so D ′ is one of the digraphs listedabove), and Aut(D) = Sm oAut(D ′).

TheoremLet Γ be a unit circulant digraph of order n. Then one of the following istrue:

1. Γ is a normal digraph of Zn,

2. Γ = Γ1 o K`, where `|n and Γ1 is a unit circulant digraph of order n/`that cannot be written as a nontrivial wreath product. ThusAut(Γ) = Aut(Γ1) o S`. Furthermore, if p | ` is prime, then p | n/` aswell,or

3. Γ = Γ1 o Kp − pΓ1, where p is prime, gcd(n/p, p) = 1, and Γ1 is a unitcirculant digraph of order n/p that cannot be written as a nontrivialwreath product. Thus Γ is a deleted wreath product, andAut(Γ) = Aut(Γ1)× Sp.

2. Γ = Γ1 o K`, where `|n and Γ1 is a unit circulant digraph of order n/`that cannot be written as a nontrivial wreath product. ThusAut(Γ) = Aut(Γ1) o S`. Furthermore, if p | ` is prime, then p | n/` aswell,

TheoremAlmost every circulant graph has automorphism group as small as possible.

It should be the case that for circulant graphs, the asymptotic behavior ofnon-normal circulant digraphs should be obtainable.

TheoremAlmost every circulant graph has automorphism group as small as possible.

It should be the case that for circulant graphs, the asymptotic behavior ofnon-normal circulant digraphs should be obtainable.

Full automorphism groups are known for

I all vertex-transitive graphs of order p (Alspach, 1973)

I all vertex-transitive graphs of order p2 (circulants by Klin andPoschel, 1978, Cayley graphs of Zp × Zp by D. and Witte, 2002)

I all vertex-transitive graphs of order pq (D. 2005, many othersinvolved)

I Zp × Zp2 , p a prime (D. )I Z3

p (D. and I. Kovacs)I circulants of odd prime-power order (Klin, Poschel (Schur rings), with

another later group theoretic proof by D.)I circulants of order a power of 2 (Klin, Najmark, and Poschel (Schur

rings))I structure theorem for all circulants (Leung, Ma, Evdomikov and

Ponomarenko 1996-2002 (Schur rings))I polynomial time algorithm for determining full automorphism groups

of all circulant digraphs (Ponomarenko, 2006)

I all vertex-transitive graphs of order p (Alspach, 1973)I all vertex-transitive graphs of order p2 (circulants by Klin and

Poschel, 1978, Cayley graphs of Zp × Zp by D. and Witte, 2002)

I all vertex-transitive graphs of order pq (D. 2005, many othersinvolved)

Poschel, 1978, Cayley graphs of Zp × Zp by D. and Witte, 2002)I all vertex-transitive graphs of order pq (D. 2005, many others

involved)

involved)I Zp × Zp2 , p a prime (D. )

I Z3p (D. and I. Kovacs)

I circulants of odd prime-power order (Klin, Poschel (Schur rings), withanother later group theoretic proof by D.)

I circulants of order a power of 2 (Klin, Najmark, and Poschel (Schurrings))

I structure theorem for all circulants (Leung, Ma, Evdomikov andPonomarenko 1996-2002 (Schur rings))

I polynomial time algorithm for determining full automorphism groupsof all circulant digraphs (Ponomarenko, 2006)

involved)I Zp × Zp2 , p a prime (D. )I Z3

p (D. and I. Kovacs)

I circulants of odd prime-power order (Klin, Poschel (Schur rings), withanother later group theoretic proof by D.)

another later group theoretic proof by D.)

rings))

Ponomarenko 1996-2002 (Schur rings))

Additional Problems

ProblemFind the full automorphism group of every vertex-transitive graph of ordera product of three not necessarily distinct primes.

ProblemSolve the isomorphism problem for every vertex-transitive graph of order aproduct of three not necessarily distinct primes.

Additional Problems

automorphism groups of cayley graphsmorris/banff-symmetries/talks/dobson-banfftalk.pdf · ted...

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