#4
Posted: Thu May 05, 2022 7:28 pm
#4
3 CAYLEY GRAPHS Let G be a group and S a generating set of G. A Cayley graph determined by G and S is a graph of the form (G, E), where there is an edge from a to b if ab-¹ € S. 1. Consider a group G as a generating set of itself. Describe the Cayley graph: what are the vertices and what are the edges. 2. Consider the group Z and your generating set S from above that has one element. Describe. the Cayley graph: what are the vertices and what are the edges; then represent the graph geometrically. 3. For each of your generating sets found earlier for Zn, Da, Da. Aa. U(16) and Z₂x Z4, describe the Cayley graph: what are the vertices and what are the edges; then represent the graph geometrically. 4. Find all the generating sets of Ss of cardinality 2. For each of these, describe the Cayley graph: what are the vertices and what are the edges; then represent the graph geometrically. 5. Find all the generating sets of Z₂ x ZxZ₂ of cardinality 3. Describe the Cayley graph: what are the vertices and what are the edges; then represent the graph geometrically. What is the shape of the diagram? 6. Consider the generating set (3,5) of Z. Describe the Cayley graph: what are the vertices and what are the edges; then represent the graph geometrically. Ente Shif End
(1,9) (el, ¹) (d, 1) (1, d) (9, 1) (1,d) (1,9) Q (1, d) (1,^) (1,6) (6,1) (d, 1) (1,9) (d, 1)
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(1,9) (el, ¹) (d, 1) (1, d) (9, 1) (1,d) (1,9) Q (1, d) (1,^) (1,6) (6,1) (d, 1) (1,9) (d, 1)