Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 2, Problem 50E
Prove the assertion made in Example 19 that the set
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Contemporary Abstract Algebra
Ch. 2 - Which of the following binary operations are...Ch. 2 - Which of the following binary operations are...Ch. 2 - Which of the following binary operations are...Ch. 2 - Which of the following sets are closed under the...Ch. 2 - In each case, find the inverse of the element...Ch. 2 - In each case, perform the indicated operation. a....Ch. 2 - Prob. 7ECh. 2 - List the elements of U(20).Ch. 2 - Show that {1, 2, 3} under multiplication modulo 4...Ch. 2 - Show that the group GL(2,R) of Example 9 is...
Ch. 2 - Let a belong to a group and a12=e . Express the...Ch. 2 - In U(9)find the inverse of 2, 7, and 8.Ch. 2 - Translate each of the following multiplicative...Ch. 2 - For group elements a, b, and c, express...Ch. 2 - Suppose that a and b belong to a group and...Ch. 2 - Show that the set {5, 15, 25, 35} is a group under...Ch. 2 - Let G be a group and let H=x1xG . Show that G=H as...Ch. 2 - List the members of K=x2xD4andL=xD4x2=e .Ch. 2 - Prove that the set of all 22 matrices with entries...Ch. 2 - For any integer n2 , show that there are at least...Ch. 2 - An abstract algebra teacher intended to give a...Ch. 2 - Let G be a group with the property that for any x,...Ch. 2 - (Law of Exponents for Abelian Groups) Let a and b...Ch. 2 - (SocksShoes Property) Draw an analogy between the...Ch. 2 - Prove that a group G is Abelian if and only if...Ch. 2 - Prove that in a group, (a1)1=a for all a.Ch. 2 - For any elements a and b from a group and any...Ch. 2 - If a1,a2,...,an belong to a group, what is the...Ch. 2 - The integers 5 and 15 are among a collection of 12...Ch. 2 - Prob. 30ECh. 2 - Prob. 31ECh. 2 - Construct a Cayley table for U(12).Ch. 2 - Suppose the table below is a group table. Fill in...Ch. 2 - Prove that in a group, (ab)2=a2b2 if and only if...Ch. 2 - Let a, b, and c be elements of a group. Solve the...Ch. 2 - Let a and b belong to a group G. Find an x in G...Ch. 2 - Let G be a finite group. Show that the number of...Ch. 2 - Give an example of a group with elements a, b, c,...Ch. 2 - Suppose that G is a group with the property that...Ch. 2 - Find an element X in D4 such that R90VXH=D .Ch. 2 - Suppose F1andF2 are distinct reflections in a...Ch. 2 - Suppose F1andF2 are distinct reflections in a...Ch. 2 - Let R be any fixed rotation and F any fixed...Ch. 2 - Let R be any fixed rotation and F any fixed...Ch. 2 - In the dihedral group Dn , let R=R360/n and let F...Ch. 2 - Prove that the set of all 33 matrices with real...Ch. 2 - Prove that if G is a group with the property that...Ch. 2 - In a finite group, show that the number of...Ch. 2 - List the six elements of GL(2,Z2) . Show that this...Ch. 2 - Prove the assertion made in Example 19 that the...Ch. 2 - Suppose that in the definition of a group G, the...Ch. 2 - Suppose that in the definition of a group G, the...
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- 9. Find all elements in each of the following groups such that . under addition. under multiplication.arrow_forward20. Let and be elements of a group . Use mathematical induction to prove each of the following statements for all positive integers . a. If the operation is multiplication, then . b. If the operation is addition and is abelian , then .arrow_forward12. 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_forward
- Use mathematical induction to prove that if a1,a2,...,an are elements of a group G, then (a1a2...an)1=an1an11...a21a11. (This is the general form of the reverse order law for inverses.)arrow_forwardTrue or False Label each of the following statements as either true or false. In a Cayley table for a group, each element appears exactly once in each row.arrow_forwardExercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.arrow_forward
- 10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .arrow_forward42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .arrow_forward
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