《Discrete Mathematics II教学-华南理工》Section 7 Generating Sets and Cayley Digraphs.pdfVIP

I.7 Generating Sets 1 Section I.7. Generating Sets and Cayley Digraphs Note. In this section, we generalize the idea of a single generator of a group to a whole set of generators of a group. Remember, a cyclic group has a single generator and is isomorphic to either Z (if it is of infinite order) or Zn (if it is of finite order), by Theorem 6.10. However, there are more groups than just the ones which are cyclic. Example 7.1. Recall the Klein 4-group, V : ∗ e a b c e e a b c a a e c b b b c e a c c b a e Then the set {a, b} is said to generate V since every element of V can be written 2 1 1 in terms of a and b: e = a , a = a , b = b , and c = ab. We can also show that V is generated by {a, c} and {b, c}. In addition, {a, b, c} is a generating set (though we could view one of the elements in this generating set as unnecessary). Exercise 7.2. Find the subgroup of Z12 generated by {4, 6}. Solution. We get all multiples of 4 and 6, so the subgroup contains 0, 4, 8, and 6. We get sums of 4 and 6: 4 + 6 = 10. Also, 2 ≡ 10 + 4 (mod 12) = 4 + 4 + 6. So the subgroup is {0, 2, 4, 6, 8, 10}. Of course, we cannot generate any odd elements of Z12. I.7 Generating Sets 2 Note. The following result goes in a little bit of a different direction in terms of subgroups. Theorem 7.4. The intersection of some subgroups Hi of a group G for i ∈ I is again a subgroup of G. (Note. Set I is called an index set for the intersec