normal and non-normal cayley graphs arising from ... · n for n 5 is a nonnormal cayley graph for...
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Normal and non-normal Cayley graphs arisingfrom generalised quadrangles
Michael Giudici
joint work with John Bamberg
Centre for the Mathematics of Symmetry and Computation
Beijing Workshop on Group Actions on CombinatorialStructures, August 2011
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Cayley graphs
• G a group
• S ⊂ G with 1 /∈ S and S = S−1
The Cayley graph Cay(G ,S) is the graph with
• vertex set G
• edges {g , sg} for all s ∈ S and g ∈ G .
Cay(G , S) is connected if and only if 〈S〉 = G .
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Automorphisms
Let G be the right regular representation of G .
Then G 6 Aut(Cay(G , S)).
Call Cay(G ,S) a graphical regular representation of G (GRR) ifequality holds.
A graph Γ is a Cayley graph for G if and only if Aut(Γ) contains aregular subgroup isomorphic to G .
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More automorphisms
Let Aut(G ,S) = {σ ∈ Aut(G ) | Sσ = S}.
Then Aut(G , S) is a group of automorphisms of Cay(G ,S).
For example, if G is abelian then
ι : G → Gg 7→ g−1
is a element of Aut(G ,S).
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Normal Cayley graphs
G o Aut(G ,S) 6 Aut(Cay(G ,S))
If equality holds, we call Cay(G ,S) a normal Cayley graph.(Mingyao Xu 1998)
Otherwise called nonnormal.
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Examples
• K4 is a normal Cayley graph for C 22 but a nonnormal Cayley
graph for C4.
• Kn for n ≥ 5 is a nonnormal Cayley graph for any group oforder n.
• Any GRR is a normal Cayley graph.
• Any Cayley graph of Cp other than Kp or pK1 is a normalCayley graph.(The full automorphism group is not 2-transitive so by atheorem of Wielandt has automorphism group contained inAGL(1, p).)
• The Hamming graph H(m, q) with q ≥ 4, is a nonnormalCayley graph for Cn
q .
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Normal and nonnormal Cayley graphs
Mingyao Xu conjectured that almost all Cayley graphs are normal.
More precisely,
minG , |G |=n
# normal Cayley graphs of G
# Cayley graphs of G→ 1 as n→∞
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A question
Yan-Quan Feng asked:
Is it possible for a graph to be both a normal and anonnormal Cayley graph for G?
Need to find Γ such that Aut(Γ) contains a normal regularsubgroup G and a nonnormal regular subgroup isomorphic to G .
Cay(G , S) is a CI-graph for G if all regular subgroups isomorphicto G are conjugate.
Li (2002) gave examples of Cayley graphs such that Aut(Γ) hastwo normal regular subgroups isomorphic to G .
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Generalised quadrangles
A generalised quadrangle is a point-line incidence geometry Q suchthat
1 any two points lie on at most one line, and
2 given a line ` and a point P not incident with `, P is collinearwith a unique point of `.
If each line is incident with s + 1 points and each point is incidentwith t + 1 lines we say that Q has order (s, t).
If s, t ≥ 2 we say Q is thick.
An automorphism of Q is a permutation of the points that mapslines to lines.
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Some graphs associated with a generalised quadrangle
Incidence graph: The bipartite graph with
• vertices the points and lines of Q• adjacency given by incidence.
Equivalent definition of a generalised quadrangle is a point-lineincidence geometry whose incidence graph has diameter 4 andgirth 8.
Has valency {t + 1, s + 1}
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Some graphs associated with a generalised quadrangle II
Collinearity graph:
• Vertices are the points of Q• two vertices are adjacent if they are collinear.
Has valency s(t + 1).
The GQ axiom implies that any clique of size at least three iscontained in the set of points on a line.
Hence any automorphism of the point graph is an automorphism ofthe GQ.
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The Classical GQ’s
Name Order Automorphism group
H(3, q2) (q2, q) PΓU(4, q)H(4, q2) (q2, q3) PΓU(5, q)W (3, q) (q, q) PΓSp(4, q)Q(4, q) (q, q) PΓO(5, q)Q−(5, q) (q, q2) PΓO−(6, q)
Take a sesquilinear or quadratic form on a vector space
• Points: totally isotropic 1-spaces
• Lines: totally isotropic 2-spaces
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Groups acting regularly on GQ’s
Ghinelli (1992): A Frobenius group or a group with nontrivialcentre cannot act regularly on a GQ of order (s, s), s even.
De Winter, K. Thas (2006): Characterised the finite thick GQswith a regular abelian group of automorphisms.
Yoshiara (2007): No GQ of order (s2, s) admits a regular group.
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p-groups
A p-group P is called special if Z (P) = P ′ = Φ(P).
A special p-group is called extraspecial if |Z (P)| = p.
Extraspecial p-groups have order p1+2n, and there are twoisomorphism classes for each order.
For p odd, one has exponent p and one has exponent p2.
Q8 and D8 are extraspecial of order 8.
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Regular groups and classical GQs
Theorem
If Q is a finite classical GQ with a regular group G ofautomorphisms then
• Q = Q−(5, 2), G extraspecial of order 27 and exponent 3.
• Q = Q−(5, 2), G extraspecial of order 27 and exponent 9.
• Q = Q−(5, 8), G ∼= GU(1, 29).9 ∼= C513 o C9.
Use classification of all regular subgroups of primitive almostsimple groups by Liebeck, Praeger and Saxl (2010)
Alternative approach independently done by De Winter, K. Thasand Shult.
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Ahrens-Szekeres-Hall quadrangles
Begin with W(3, q) defined by the alternating form
β(x , y) := x1y4 − y1x4 + x2y3 − y2x3
Let x = 〈(1, 0, 0, 0)〉.
Define a new GQ, Qx, with
• points: the points of W(3, q) not collinear with x , that is, all〈(a, b, c , 1)〉
• lines:
(a) lines of W (3, q) not containing x , and(b) the hyperbolic 2-spaces containing x .
Qx is a GQ of order (q − 1, q + 1).
This construction is known as Payne derivation.
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Automorphisms
• PΓSp(4, q)x 6 Aut(Qx)
• Grundhofer, Joswig, Stroppel (1994): Equality holds forq ≥ 5.
• Note Sp(4, q)x consists of all matrices λ 0 0uT A 0z v λ−1
with
A ∈ GL(2, q) such that AJAT = J,
z ∈ GF(q) and u, v ∈ GF(q)2 such that u = λvJAT
where J =(
0 1−1 0
).
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An obvious regular subgroup
Let
E =
1 0 0 0−c 1 0 0b 0 1 0a b c 1
∣∣∣ a, b, c ∈ GF(q)
C ΓSp(4, q)x
• |E | = q3,
• acts regularly on the points of Qx,
• elementary abelian for q even,
• special of exponent p for q odd (Heisenberg group).
So the collinearity graph of one of these GQs is a normal Cayleygraph for q ≥ 5.
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But there are more . . .
q # conjugacy classes # isomorphism classes
3 2 2 (this is Q−(5, 2))4 58 305 2 17 2 18 14 89 5 5
11 2 113 2 116 231 -17 2 119 2 123 2 125 7 -
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An element
For α ∈ GF(q), let
θα =
1 0 0 0−α 1 0 0−α2 α 1 0
0 0 α 1
∈ Sp(4, q)x.
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Construction 1
Let {α1, . . . , αf } be a basis for GF(q) over GF(p).
P =
⟨1 0 0 00 1 0 0b 0 1 0a b 0 1
, θα1 , . . . , θαf
⟩
• P acts regularly on points and is not normal in Aut(Qx)
• nonabelian for q > 2
• not special for q even, special for q odd
• exponent 9 for p = 3
• P ∼= E for p ≥ 5
• So for p ≥ 5 the collinearity graph is both a normal andnonnormal Cayley graph for isomorphic groups.
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Some other examples
• Halved folded 8-cube for C 62 (Royle 2008)
• a Cayley graph for C 23 oC 2
2 of valency 14 (Giudici-Smith 2010)
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Construction 2
Let U ⊕W be a decomposition of GF(q), q = pf , f ≥ 2.
Let {α1, . . . , αk} be a basis for U over GF(p).
SU,W =
⟨1 0 0 00 1 0 0b 0 1 0a b 0 1
,
1 0 0 0−w 1 0 0
0 0 1 00 0 w 1
, θα1 , . . . , θαk
⟩
• SU,W acts regularly on points,
• for U a 1-space:• E 6∼= SU,W 6∼= P• SU,W is not special
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Significance for GQs
The class of groups that can act regularly on the set of points of aGQ is richer/wilder than previously thought.
Such a group can be
• a nonabelian 2-group,
• a 2-group of nilpotency class 7,
• p-groups that are not Heisenberg groups,
• p-groups that are not special.