Automorphism Groups of Cayley Graphs Ted Dobson Department of Mathematics & Statistics Mississippi State University [email protected] http://www2.msstate.edu/∼dobson/ Symmetries of Graphs and Networks November 23{28, 2008 Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs A circulant graph of order n is simply a Cayley graph of Zn. Basic Definitions Definition Let G be a group and S ⊂ G. We define the Cayley digraph of G with connection set S to be the digraph Γ = Γ(G; S) defined by V (Γ) = G and E(Γ) = f(g; gs): s 2 S; g 2 Gg. If S = S−1, then Γ is a Cayley graph of G with connection set S. Note that GL = fx ! gx : g 2 Gg ≤ Aut(Γ). Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Basic Definitions Definition Let G be a group and S ⊂ G. We define the Cayley digraph of G with connection set S to be the digraph Γ = Γ(G; S) defined by V (Γ) = G and E(Γ) = f(g; gs): s 2 S; g 2 Gg. If S = S−1, then Γ is a Cayley graph of G with connection set S. Note that GL = fx ! gx : g 2 Gg ≤ Aut(Γ). A circulant graph of order n is simply a Cayley graph of Zn. Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Observe that every vertex-transitive graph Γ of prime order is isomorphic to a circulant graph of prime order as Aut(Γ) contains a subgroup of order p, which, after a relabeling, we may assume is h(0; 1;:::; p − 1)i = hx ! x + 1i = (Zp)L. So circulant! Theorem (Burnside, 1901) Let G ≤ Sp, p a prime, contain (Zp)L. Then ∗ G ≤ AGL(1; p) = fx ! ax + b : a 2 Zp; b 2 Zpg or G is doubly-transitive. Note that if G ≤ Aut(Γ) is doubly-transitive, then Aut(Γ) = SX and Γ is complete or has no edges. Automorphisms of prime order circulants Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem (Burnside, 1901) Let G ≤ Sp, p a prime, contain (Zp)L. Then ∗ G ≤ AGL(1; p) = fx ! ax + b : a 2 Zp; b 2 Zpg or G is doubly-transitive. Note that if G ≤ Aut(Γ) is doubly-transitive, then Aut(Γ) = SX and Γ is complete or has no edges. Automorphisms of prime order circulants Observe that every vertex-transitive graph Γ of prime order is isomorphic to a circulant graph of prime order as Aut(Γ) contains a subgroup of order p, which, after a relabeling, we may assume is h(0; 1;:::; p − 1)i = hx ! x + 1i = (Zp)L. So circulant! Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Note that if G ≤ Aut(Γ) is doubly-transitive, then Aut(Γ) = SX and Γ is complete or has no edges. Automorphisms of prime order circulants Observe that every vertex-transitive graph Γ of prime order is isomorphic to a circulant graph of prime order as Aut(Γ) contains a subgroup of order p, which, after a relabeling, we may assume is h(0; 1;:::; p − 1)i = hx ! x + 1i = (Zp)L. So circulant! Theorem (Burnside, 1901) Let G ≤ Sp, p a prime, contain (Zp)L. Then ∗ G ≤ AGL(1; p) = fx ! ax + b : a 2 Zp; b 2 Zpg or G is doubly-transitive. Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Automorphisms of prime order circulants Observe that every vertex-transitive graph Γ of prime order is isomorphic to a circulant graph of prime order as Aut(Γ) contains a subgroup of order p, which, after a relabeling, we may assume is h(0; 1;:::; p − 1)i = hx ! x + 1i = (Zp)L. So circulant! Theorem (Burnside, 1901) Let G ≤ Sp, p a prime, contain (Zp)L. Then ∗ G ≤ AGL(1; p) = fx ! ax + b : a 2 Zp; b 2 Zpg or G is doubly-transitive. Note that if G ≤ Aut(Γ) is doubly-transitive, then Aut(Γ) = SX and Γ is complete or has no edges. Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs B. Alspach (1973) first observed this, and went on to explicitly enumerate the vertex-transitive graphs of prime order. Burnside's Theorem is equivalent to Theorem Let G ≤ Sp, p a prime, be transitive. Then either G contains a normal Sylow p-subgroup or G is doubly-transitive. This gives an algorithm for determining the full automorphism group of a circulant graph Γ = Γ(Zp; S). Note that x ! x + b is always contained in ∗ Aut(Γ), so we need only check which a 2 Zp satisfy a · S = fas : s 2 Sg = S (we observe that AGL(1; p) is itself doubly-transitive, so if all such x ! ax are in Aut(Γ), then Aut(Γ) = Sp). Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Burnside's Theorem is equivalent to Theorem Let G ≤ Sp, p a prime, be transitive. Then either G contains a normal Sylow p-subgroup or G is doubly-transitive. This gives an algorithm for determining the full automorphism group of a circulant graph Γ = Γ(Zp; S). Note that x ! x + b is always contained in ∗ Aut(Γ), so we need only check which a 2 Zp satisfy a · S = fas : s 2 Sg = S (we observe that AGL(1; p) is itself doubly-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 enumerate the vertex-transitive graphs of prime order. Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let G ≤ Sp, p a prime, be transitive. Then either G contains a normal Sylow p-subgroup or G is doubly-transitive. This gives an algorithm for determining the full automorphism group of a circulant graph Γ = Γ(Zp; S). Note that x ! x + b is always contained in ∗ Aut(Γ), so we need only check which a 2 Zp satisfy a · S = fas : s 2 Sg = S (we observe that AGL(1; p) is itself doubly-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 enumerate the vertex-transitive graphs of prime order. Burnside's Theorem is equivalent to Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs This gives an algorithm for determining the full automorphism group of a circulant graph Γ = Γ(Zp; S). Note that x ! x + b is always contained in ∗ Aut(Γ), so we need only check which a 2 Zp satisfy a · S = fas : s 2 Sg = S (we observe that AGL(1; p) is itself doubly-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 enumerate the vertex-transitive graphs of prime order. Burnside's Theorem is equivalent to Theorem Let G ≤ Sp, p a prime, be transitive. Then either G contains a normal Sylow p-subgroup or G is doubly-transitive. Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem (D., D. Witte, 2002) There are exactly 2p − 1 transitive p-subgroups P of Sp2 up to conjugation, and all but three have the property that if G ≤ Sp2 with Sylow 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 that every minimal transitive subgroup of P is cyclic. If G ≤ Spk with Sylow p-subgroup P, then either P/G or G is doubly-transitive. It is probable that many more generalizations are true, but it also seems likely that \most classes" of automorphism groups of graphs cannot be obtained in this way. Can Burnside's Theorem be generalized? Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem (D., 2005) Let P be a transitive p-subgroup of Spk , p an odd prime, k ≥ 1, such that every minimal transitive subgroup of P is cyclic. If G ≤ Spk with Sylow p-subgroup P, then either P/G or G is doubly-transitive. It is probable that many more generalizations are true, but it also seems likely that \most classes" of automorphism groups of graphs cannot be obtained in this way. Can Burnside's Theorem be generalized? Theorem (D., D. Witte, 2002) There are exactly 2p − 1 transitive p-subgroups P of Sp2 up to conjugation, and all but three have the property that if G ≤ Sp2 with Sylow p-subgroup P, then either P/G or G is doubly-transitive. Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs It is probable that many more generalizations are true, but it also seems likely that \most classes" of automorphism groups of graphs cannot be obtained in this way. Can Burnside's Theorem be generalized? Theorem (D., D. Witte, 2002) There are exactly 2p − 1 transitive p-subgroups P of Sp2 up to conjugation, and all but three have the property that if G ≤ Sp2 with Sylow 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 that every minimal transitive subgroup of P is cyclic. If G ≤ Spk with Sylow p-subgroup P, then either P/G or G is doubly-transitive. Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Can Burnside's Theorem be generalized? Theorem (D., D.