A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Chapter 5.2, Problem 2E
Given that
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A Transition to Advanced Mathematics
Ch. 5.1 - The Cayley tables for operations o,*,+, and are...Ch. 5.1 - Prob. 2ECh. 5.1 - Prob. 3ECh. 5.1 - Prob. 4ECh. 5.1 - Give an example of an algebraic structure of order...Ch. 5.1 - Prob. 6ECh. 5.1 - Show that the structure ({1},), with operation ...Ch. 5.1 - (a)In the group G of Exercise 2, find x such that...Ch. 5.1 - Show that (,), with operation # defined by...Ch. 5.1 - Construct the operation table for each of the...
Ch. 5.1 - Prob. 11ECh. 5.1 - (a)Prove that (m,+) is associative and commutative...Ch. 5.1 - Suppose m and m2. Prove that 1 and m1 are distinct...Ch. 5.1 - Let m and a be natural numbers with am. Complete...Ch. 5.1 - Prob. 15ECh. 5.1 - Prob. 16ECh. 5.1 - Prob. 17ECh. 5.1 - Consider the set A={a,b,c,d} with operation ogiven...Ch. 5.1 - Repeat Exercise 2 with the operation * given by...Ch. 5.1 - Let m,n and M=A:A is an mn matrix with real number...Ch. 5.1 - Prob. 21ECh. 5.1 - Prob. 22ECh. 5.2 - Show that each of the following algebraic...Ch. 5.2 - Given that G={e,u,v,w} is a group of order 4 with...Ch. 5.2 - Prob. 3ECh. 5.2 - Give an example of an algebraic system (G,o) that...Ch. 5.2 - Construct the operation table for S2. Is S2...Ch. 5.2 - Prob. 6ECh. 5.2 - Let G be a group and aiG for all n. Prove that...Ch. 5.2 - Prove part (d) of Theorem 6.2.3. That is, prove...Ch. 5.2 - Prob. 9ECh. 5.2 - Prob. 10ECh. 5.2 - Prob. 11ECh. 5.2 - Assign a grade of A (correct), C (partially...Ch. 5.3 - Assign a grade of A (correct), C (partially...Ch. 5.3 - Find all subgroups of (8,+). (U11,). (5,+). (U7,)....Ch. 5.3 - In the group S4, find two different subgroups that...Ch. 5.3 - Prove that if G is a group and H is a subgroup of...Ch. 5.3 - Prove that if H and K are subgroups of a group G,...Ch. 5.3 - Let G be a group and H be a subgroup of G. If H is...Ch. 5.3 - Prob. 7ECh. 5.3 - Prob. 8ECh. 5.3 - Prob. 9ECh. 5.3 - List all generators of each cyclic group in...Ch. 5.3 - Prob. 11ECh. 5.3 - Let G be a group, and let H be a subgroup of G....Ch. 5.3 - Let ({0},) be the group of nonzero complex numbers...Ch. 5.3 - Prob. 14ECh. 5.3 - Prob. 15ECh. 5.3 - Let G=a be a cyclic group of order 30. What is the...Ch. 5.4 - Is S3 isomorphic to (6,+)? Explain.Ch. 5.4 - Prob. 2ECh. 5.4 - Use the method of proof of Cayley's Theorem to...Ch. 5.4 - Define f:++ by f(x)=x where + is the set of all...Ch. 5.4 - Assign a grade of A (correct), C (partially...Ch. 5.4 - Prob. 6ECh. 5.4 - Define on by setting (a,b)(c,d)=(acbd,ad+bc)....Ch. 5.4 - Let f the set of all real-valued integrable...Ch. 5.4 - Prob. 9ECh. 5.4 - Find the order of each element of the group S3....Ch. 5.4 - Prob. 11ECh. 5.4 - Let (3,+) and (6,+) be the groups in Exercise 10,...Ch. 5.4 - Prob. 13ECh. 5.4 - Prob. 14ECh. 5.4 - Prob. 15ECh. 5.4 - Prob. 16ECh. 5.4 - Prob. 17ECh. 5.5 - Prob. 1ECh. 5.5 - Prob. 2ECh. 5.5 - Show that any two groups of order 2 are...Ch. 5.5 - Show that the function h: defined by h(x)=3x is...Ch. 5.5 - Let R be the equivalence relation on ({0}) given...Ch. 5.5 - Prob. 6ECh. 5.5 - Prob. 7ECh. 5.5 - Let (R,+,) be an algebraic structure such that...Ch. 5.5 - Assign a grade of A (correct), C (partially...Ch. 5.5 - Let M be the set of all 22 matrices with real...
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- If a is an element of order m in a group G and ak=e, prove that m divides k.arrow_forwardConstruct a multiplication table for the group D5 of rigid motions of a regular pentagon with vertices 1,2,3,4,5.arrow_forwardLet A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)arrow_forward
- 24. Let be a group and its center. Prove or disprove that if is in, then and are in.arrow_forward45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )arrow_forwardShow that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.arrow_forward
- Construct a multiplication table for the group G of rigid motions of a rectangle with vertices 1,2,3,4 if the rectangle is not a square.arrow_forwardExercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forwardIn Exercises 1- 9, let be the given group. Write out the elements of a group of permutations that is isomorphic to, and exhibit an isomorphism from to this group. 9. Let be the octic group .arrow_forward
- 12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.arrow_forwardIn Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let be the subgroup of the octic group . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of . Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group of rigid motions of a square The elements of the group are as follows: 1. the identity mapping 2. the counterclockwise rotation through about the center 3. the counterclockwise rotation through about the center 4. the counterclockwise rotation through about the center 5. the reflection about the horizontal line 6. the reflection about the diagonal 7. the reflection about the vertical line 8. the reflection about the diagonal . The dihedral group of rigid motions of the square is also known as the octic group. The multiplication table for is requested in Exercise 20 of this section.arrow_forwardIn Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.arrow_forward
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