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Pretty Pictures of Cayley Graphs Pretty Pictures of Cayley Graphs Sara Fish Caltech Math Club Friday, February 7, 2020∗ I apologize for typos or errors. Feel free to contact me if you spot any. Abstract The Cayley graph of a (finitely generated) group G is a directed edge-colored graph which encodes the structure of G. In this talk, we define Cayley graphs, visualize various properties of groups using Cayley graphs, and most importantly, look at lots of pretty examples. Contents 1 Cayley graphs 1 2 Group actions on Cayley graphs 3 3 Subgroups in Cayley graphs 4 4 Normal subgroups in Cayley graphs 5 5 Direct and semidirect products 6 5.1 Cayley graphs of direct products . .7 5.2 Cayley graphs of semidirect products . .8 6 Pretty pictures 9 1 Cayley graphs Let G be a group. Definition 1. A subset S ⊆ G is a generating set of G if for every g 2 G, there exist s1; : : : ; sn 2 S such that G = s1 : : : sn. We write G = hSi. We say G is finitely generated if there exists a finite generating set of G. Example 1. 1. Z6 is generated by h1i and h2; 3i, but not h2i. ∗These notes heavily borrow from notes I wrote in 2018, for a two-day Mathcamp class on this topic. See www.sarafish.com. 1 1 CAYLEY GRAPHS 2. D8 = hr; si. 3. Sn = hall 2-cyclesi = h(123 : : : n); (12)i. 4. Trivially, G = hGi. Of course, we care about small generating sets which say something interesting about the group. 5. If G infinite, it may still be finitely generated. F2 = ha; bi, the free group with two generators. Another example is Z, generated by f1g, but also any finite set of integers with no common prime divisor. 6. A group need not be finitely generated: take for example (Q; +). Next we define the notion of a Cayley graph. Roughly speaking, the Cayley graph of a group G is useful because it encodes various properties of G, such as the structure of its subgroups. Definition 2. Let G be a group with finite generating set S. The Cayley graph Γ(G; S) is a directed, edge-colored graph satisfying the following. • V = G (Each vertex is labeled with its group element.) • E = f(g; gs); 8g 2 G; s 2 Sg (For every g and every s, draw an edge going from g to gs.) • Color each edge (g; gs) with the color s. Example 2. 1. Z6, with generating set f1g. 0 5 1 4 2 3 2. Z6, with generating set f2; 3g. 0 0 2 4 5 1 4 2 3 5 1 3 3. D8, with generating set fr; sg. 2 2 GROUP ACTIONS ON CAYLEY GRAPHS r3s r2s r3 r2 e r s rs 2 Group actions on Cayley graphs Recall that a group G can act on itself (on the left) as follows: Each g is associated with some φg 2 Aut(G), where φg(x) = gx for all x 2 G. The action of G on itself induces an action of G on Γ(G; S). Definition 3. 1. Let Γ = (V; E) be a graph. A graph automorphism is a bijection φ : V ! V which preserves edges. That is, (v; w) 2 E if and only if (φ(v); φ(w)) 2 E. 2. If Γ is edge-colored and directed (as with Cayley graphs), we also require that φ preserve the color and direction edges. That is, (v; w) and (φ(v); φ(w)) must also have the same color and direction. Example 3. has 4! = 24 graph automorphisms. has 2 · 2 = 4 Cayley graph automorphisms (preserve color and direction). Definition 4. Let Γ(G; S) be a Cayley graph. For each a 2 G, let φa : V ! V where: for all g 2 G, φa(g) = ag. In this way, G acts on its Cayley graph Γ, by permuting the labels of the vertices. Proposition 1. The action of any g 2 G on Γ (via the automorphism φg) preserves edges. Short \proof". Let (v; vs) be an edge. After a 2 G acts on Γ, (v; vs) becomes (av; a(vs)) = (av; (av)s). Example 4. 1. Consider the Cayley graph of Z6, with generator h1i. The only graph automorphisms are the six rotations, which correspond to the group elements. 3 3 SUBGROUPS IN CAYLEY GRAPHS 2. Consider the Cayley graph of Z6, with generating set h2; 3i. The graph automorphisms are still the six rotations, and these preserve all of the edges. Example 5. The element r is acting on Γ(D8; fr; sg). 3 r3s r2s s r s 3 r3 r2 e r e r r r2 s rs rs r2s Remark 1. When following along an edge s of a Cayley graph, we multiply on the right, that is, g 7! gs. When an element a acts on a Cayley graph, we multiply on the left, that is, g 7! ag. It is crucial that these two things are from opposite sides! In summary: Take a group element g 2 G, and multiply each vertex label in Γ(G; S) by g on the left. The edge relations will be preserved. A consequence of this is that Cayley graphs \look the same" at every vertex. 3 Subgroups in Cayley graphs Example 6. Take a Cayley graph Γ, and consider what happens when we remove all edges of a particular color. 1. Take Γ(Z6; f2; 3g). If we remove the 2 edges, we are left with three line segments, correspond- ing to the three cosets of the subgroup h3i. If we remove the 3 edges, we are left with two triangles, corresponding to the two cosets of the subgroup h2i. 2. Take Γ(D8; fr; sg). If we remove the r edges, we are left with four cosets of hsi. If we remove the s edges, we are left with two cosets of hri. 4 4 NORMAL SUBGROUPS IN CAYLEY GRAPHS r3s r2s r3s r2s r3 r2 r3 r2 remove s edges Inner square: hri −! Outer square: shri e r e r s rs s rs r3s r2s r3s r2s 3 2 3 2 r r r r These correspond to remove r edges r3hsi r2hsi −! hsi rhsi e r e r s rs s rs Let's explain this phenomenon. Proposition 2. Let G be a group and H ≤ G. Suppose H = hS0i and G = hSi, where S0 ⊆ S. Then, removing all edges colored by S nS0 leaves [G : H] copies of Γ(H; S0). Each copy corresponds to a left coset gH. Proof. Sketch. 1. Consider some g 2 G. Using edges in S0, we can reach everything in gH. (Draw [G : H] identical blobs, which are the cosets gH. Draw edges within these blobs { these are the edges in S0.) 2. For all g1; g2 2 G, g1H and g2H look the same, because there is a graph automorphism φ −1 g2g1 between them. (Draw dotted edges between the blobs. These are the edges in S n S0.) 4 Normal subgroups in Cayley graphs Definition 5. A subgroup N ≤ G is normal if and only if for all g 2 G, we have gN = Ng. We write N E G. Proposition 3. Let N ≤ G. Then N E G if and only if: If you take any s 2 S, moving each element N along the s edge gives a left coset gN. Example 7. Testing whether subgroups of Γ(D8; fr; sg) are normal: 5 5 DIRECT AND SEMIDIRECT PRODUCTS Subgroup r3s r2s move hri along r Cosets are −! r3 r2 hri e r move hri along s −! so hri is normal. s rs 3 2 r s r s Cosets are r3 r2 move hsi along r hsi −! e r so hsi is s rs NOT normal. 3 2 r s r s Cosets are the move hr2i along r −! four \diagonals" r3 r2 hr2i e r move hr2i along s −! 2 s rs so hr i is normal. 5 Direct and semidirect products Recall that direct and semidirect products give two ways of building larger groups from known smaller groups. We can do this to Cayley graphs: given the Cayley graphs of two smaller groups N and H, we can combine these Cayley graphs (with some modifications) to get the Cayley graphs of N × H and N oΦ H. First, let's recall the definitions of direct and semidirect products. Definition 6. Let N and H be groups. The direct product of N and H, denoted N × H, is the set N × H with multiplication rule (n1; h1) · (n2; h2) = (n1n2; h1h2): Definition 7. Let N and H be groups. Let Φ : H ! Aut(N) be a homomorphism (describing how to conjugate each element of N by each element of H). The semidirect product of N and 6 5.1 Cayley graphs of direct products 5 DIRECT AND SEMIDIRECT PRODUCTS H, denoted N oΦ H, is the set N × H with multiplication rule (n1; h1) · (n2; h2) = (n1Φ(h1)(n2); h1h2): Finally, set N = hSi and H = hT i. 5.1 Cayley graphs of direct products Example 8. Consider the Cayley graph of Z4 × Z2. Note that it can be seen as 2 copies of Z4 stitched together by copies of Z2. Or it can be seen as 4 copies of Z2 stitched together by copies of Z4. −! −! Z2 Z4 Z2 × Z4 Z4 × Z2 Put a copy of Z4 at each vertex of Z2. Put a copy of Z2 at each vertex of Z4.
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