Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 2, Problem 23E
(Law of Exponents for Abelian Groups) Let a and b be elements of an Abelian group and let n be any integer. Show that
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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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- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.arrow_forward27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.arrow_forwardIn Example 3, the group S(A) is nonabelian where A={ 1,2,3 }. Exhibit a set A such that S(A) is abelian. Example 3. We shall take A={ 1,2,3 } and obtain an explicit example of S(A). In order to define an element f of S(A), we need to specify f(1), f(2), and f(3). There are three possible choices for f(1). Since f is to be bijective, there are two choices for f(2) after f(1) has been designated, and then only once choice for f(3). Hence there are 3!=321 different mappings f in S(A).arrow_forward
- 11. Show that is a generating set for the additive abelian group if and only ifarrow_forward17. Let and be elements of a group. Prove that is abelian if and only if .arrow_forwardSuppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.arrow_forward
- 23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .arrow_forward15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .arrow_forward9. Find all homomorphic images of the octic group.arrow_forward
- Find two groups of order 6 that are not isomorphic.arrow_forwardProve that the Cartesian product 24 is an abelian group with respect to the binary operation of addition as defined in Example 11. (Sec. 3.4,27b, Sec. 5.1,53,) Example 11. Consider the additive groups 2 and 4. To avoid any unnecessary confusion we write [ a ]2 and [ a ]4 to designate elements in 2 and 4, respectively. The Cartesian product of 2 and 4 can be expressed as 24={ ([ a ]2,[ b ]4)[ a ]22,[ b ]44 } Sec. 3.4,27b 27. Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 5.1,53 53. Rework Exercise 52 with the direct sum 24.arrow_forward6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.arrow_forward
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