Problem 4. Let G be a group, and suppose that g € G satisfies |g| = n < ∞. Find the order of the element gk for any k € Z+. Hint: |gk| will depend on k and n.
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- Exercises 8. Let be an element of order in a group. Find the order of each of the following. a. b. c. d. e. f. g. h.Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.
- For each of the following values of n, find all distinct generators of the group Un described in Exercise 11. a. n=7 b. n=5 c. n=11 d. n=13 e. n=17 f. n=19Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?22. Find the center for each of the following groups . a. in Exercise 34 of section 3.1. b. in Exercise 36 of section 3.1. c. in Exercise 35 of section 3.1. d., the general linear group of order over. Exercise 34 of section 3.1. Let be the set of eight elements with identity element and noncommutative multiplication given by for all in (The circular order of multiplication is indicated by the diagram in Figure .) Given that is a group of order , write out the multiplication table for . This group is known as the quaternion group. Exercise 36 of section 3.1 Consider the matrices in , and let . Given that is a group of order 8 with respect to multiplication, write out a multiplication table for. Exercise 35 of section 3.1. A permutation matrix is a matrix that can be obtained from an identity matrix by interchanging the rows one or more times (that is, by permuting the rows). For the permutation matrices are and the five matrices. Given that is a group of order with respect to matrix multiplication, write out a multiplication table for .
- Exercises 9. For each of the following values of, find all distinct generators of the cyclic group under addition. a. b. c. d. e. f.Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.Exercises 3. Find the order of each element of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .
- Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .1. Let G and H be groups, and let g € G and h & H. Find a formula for the order of the element (g, h) = G × H in terms of the order of g in G and the order of h in H.4.1.7. Let G = Dg act on N = D3 by conjugation. Make an 8 x 8 table, where the rows are indexed by the elements of G and the columns are indexed 1.2. The Cayley Graph of a Group Action* 91 by the elements of 2. The entry in row g and column x is the result of the action of g on x (that is g · x.) Complete the table. (a) In each row do you get every element of 2? Does any element of N ever appear in any row more than once? Is this a coincidence? (b) Does an element of 2 ever appear in any column more than once? Is it possible for a column to contain only one element of ? When does this happen? (c) Can two rows be identical? (d) Is it possible to tell the order of the elements of the group from this table?
