Normal and non-normal Cayley graphs arising from generalised quadrangles Michael Giudici joint work with John Bamberg Centre for the Mathematics of Symmetry and Computation Beijing Workshop on Group Actions on Combinatorial Structures, August 2011
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.
Automorphisms Let Ĝ be the right regular representation of G. Then Ĝ Aut(Cay(G, S)). Call Cay(G, S) a graphical regular representation of G (GRR) if equality holds. A graph Γ is a Cayley graph for G if and only if Aut(Γ) contains a regular subgroup isomorphic to G.
More automorphisms Let Aut(G, S) = {σ Aut(G) S σ = S}. Then Aut(G, S) is a group of automorphisms of Cay(G, S). For example, if G is abelian then is a element of Aut(G, S). ι : G G g g 1
Normal Cayley graphs Ĝ Aut(G, S) Aut(Cay(G, S)) If equality holds, we call Cay(G, S) a normal Cayley graph. (Mingyao Xu 1998) Otherwise called nonnormal.
Examples K 4 is a normal Cayley graph for C2 2 but a nonnormal Cayley graph for C 4. K n for n 5 is a nonnormal Cayley graph for any group of order n. Any GRR is a normal Cayley graph. Any Cayley graph of C p other than K p or pk 1 is a normal Cayley graph. (The full automorphism group is not 2-transitive so by a theorem of Wielandt has automorphism group contained in AGL(1, p).) The Hamming graph H(m, q) with q 4, is a nonnormal Cayley graph for C n q.
Normal and nonnormal Cayley graphs Mingyao Xu conjectured that almost all Cayley graphs are normal. More precisely, # normal Cayley graphs of G min G, G =n # Cayley graphs of G 1 as n
A question Yan-Quan Feng asked: Is it possible for a graph to be both a normal and a nonnormal Cayley graph for G? Need to find Γ such that Aut(Γ) contains a normal regular subgroup G and a nonnormal regular subgroup isomorphic to G. Cay(G, S) is a CI-graph for G if all regular subgroups isomorphic to G are conjugate. Li (2002) gave examples of Cayley graphs such that Aut(Γ) has two normal regular subgroups isomorphic to G.
Generalised quadrangles A generalised quadrangle is a point-line incidence geometry Q such that 1 any two points lie on at most one line, and 2 given a line l and a point P not incident with l, P is collinear with a unique point of l. If each line is incident with s + 1 points and each point is incident with 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 maps lines to lines.
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-line incidence geometry whose incidence graph has diameter 4 and girth 8. Has valency {t + 1, s + 1}
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 is contained in the set of points on a line. Hence any automorphism of the point graph is an automorphism of the GQ.
The Classical GQ s Name Order Automorphism group H(3, q 2 ) (q 2, q) PΓU(4, q) H(4, q 2 ) (q 2, q 3 ) PΓU(5, q) W (3, q) (q, q) PΓSp(4, q) Q(4, q) (q, q) PΓO(5, q) Q (5, q) (q, q 2 ) PΓO (6, q) Take a sesquilinear or quadratic form on a vector space Points: totally isotropic 1-spaces Lines: totally isotropic 2-spaces
Groups acting regularly on GQ s Ghinelli (1992): A Frobenius group or a group with nontrivial centre cannot act regularly on a GQ of order (s, s), s even. De Winter, K. Thas (2006): Characterised the finite thick GQs with a regular abelian group of automorphisms. Yoshiara (2007): No GQ of order (s 2, s) admits a regular group.
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 p 1+2n, and there are two isomorphism classes for each order. For p odd, one has exponent p and one has exponent p 2. Q 8 and D 8 are extraspecial of order 8.
Regular groups and classical GQs Theorem If Q is a finite classical GQ with a regular group G of automorphisms 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, 2 9 ).9 = C 513 C 9. Use classification of all regular subgroups of primitive almost simple groups by Liebeck, Praeger and Saxl (2010) Alternative approach independently done by De Winter, K. Thas and Shult.
Ahrens-Szekeres-Hall quadrangles Begin with W(3, q) defined by the alternating form β(x, y) := x 1 y 4 y 1 x 4 + x 2 y 3 y 2 x 3 Let x = (1, 0, 0, 0). Define a new GQ, Q x, 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. Q x is a GQ of order (q 1, q + 1). This construction is known as Payne derivation.
Automorphisms PΓSp(4, q) x Aut(Q x ) Grundhöfer, Joswig, Stroppel (1994): Equality holds for q 5. Note Sp(4, q) x consists of all matrices with λ 0 0 u A 0 z v λ 1 A GL(2, q) such that AJA T = J, z GF(q) and u, v GF(q) 2 such that u = λvja T where J = ( 0 1 1 0 ).
An obvious regular subgroup Let 1 0 0 0 E = c 1 0 0 b 0 1 0 a, b, c GF(q) ΓSp(4, q) x a b c 1 E = q 3, acts regularly on the points of Q x, 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 Cayley graph for q 5.
But there are more... q # conjugacy classes # isomorphism classes 3 2 2 (this is Q (5, 2)) 4 58 30 5 2 1 7 2 1 8 14 8 9 5 5 11 2 1 13 2 1 16 231-17 2 1 19 2 1 23 2 1 25 7 -
An element For α GF(q), let 1 0 0 0 θ α = α 1 0 0 α 2 α 1 0 Sp(4, q) x. 0 0 α 1
Construction 1 Let {α 1,..., α f } be a basis for GF(q) over GF(p). 1 0 0 0 P = 0 1 0 0 b 0 1 0, θ α 1,..., θ αf a b 0 1 P acts regularly on points and is not normal in Aut(Q x ) 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 and nonnormal Cayley graph for isomorphic groups.
Some other examples Halved folded 8-cube for C2 6 (Royle 2008) a Cayley graph for C3 2 C 2 2 of valency 14 (Giudici-Smith 2010)
Construction 2 Let U W be a decomposition of GF(q), q = p f, f 2. Let {α 1,..., α k } be a basis for U over GF(p). 1 0 0 0 1 0 0 0 S U,W = 0 1 0 0 b 0 1 0, w 1 0 0 0 0 1 0, θ α 1,..., θ αk a b 0 1 0 0 w 1 S U,W acts regularly on points, for U a 1-space: E = SU,W = P S U,W is not special
Significance for GQs The class of groups that can act regularly on the set of points of a GQ 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.