Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 10, Problem 6E
Let G be the group of all polynomials with real coefficients under addition.For each f in G, let
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Contemporary Abstract Algebra
Ch. 10 - Let R* be the group of nonzero real numbers under...Ch. 10 - Let G be the group of all polynomials with real...Ch. 10 - Prob. 7ECh. 10 - Explain why the correspondence x3x from Z12toZ10...Ch. 10 - Prob. 15ECh. 10 - Prove that there is no homomorphism from...Ch. 10 - Let be a homomorphism from a finite group G to G...Ch. 10 - Prob. 39ECh. 10 - Show that a homomorphism defined on a cyclic group...Ch. 10 - Suppose there is a homomorphism from G onto Z2Z2...
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- Consider the additive group of real numbers. Let be a mapping from to , where equality and addition are defined in Exercise 52 of Section 3.1. Prove or disprove that each of the following mappings is a homomorphism. If is a homomorphism, find ker , and decide whether is an epimorphism or a monomorphism. a. (x,y)=xy b. (x,y)=2xarrow_forwardExercises 22. Let be a finite cyclic group of order with generators and . Prove that the mapping is an automorphism of .arrow_forwardLet G be a group. Prove that the relation R on G, defined by xRy if and only if there exist an aG such that y=a1xa, is an equivalence relation. Let xG. Find [ x ], the equivalence class containing x, if G is abelian. (Sec 3.3,23) Sec. 3.3, #23: 23. Let R be the equivalence relation on G defined by xRy if and only if there exists an element a in G such that y=a1xa. If x(G), find [ x ], the equivalence class containing x.arrow_forward
- 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_forward24. The center of a group is defined as Prove that is a normal subgroup of .arrow_forwardEach of the following rules determines a mapping :GG, where G is the group of all nonzero real numbers under multiplication. Decide in each case whether or not is an endomorphism. For those that are endomorphisms, state the kernel and decide whether is an epimorphism or a monomorphism. a. (x)=| x | b. (x)=1x c. (x)=x d. (x)=x2 e. (x)=| x |x f. (x)=x2+1 g. (x)=x3 h. (x)=x2arrow_forward
- Let H be a torsion subgroup of an abelian group G. That is, H is the set of all elements of finite order in G. Prove that H is normal in G.arrow_forwardConsider the additive group of real numbers. Prove or disprove that each of the following mappings : is an automorphism. Equality and addition are defined on in Exercise 52 of section 3.1. a. (x,y)=(y,x) b. (x,y)=(x,y) Sec. 3.1,52 Let G1 and G2 be groups with respect to addition. Define equality and addition in the Cartesian product by G1G2 (a,b)=(a,b) if and only if a=a and b=a (a,b)+(c,d)=(ac,bd) Where indicates the addition in G1 and indicates the addition in G2. Prove that G1G2 is a group with respect to addition. Prove that G1G2 is abelian if both G1 and G2 are abelian. For notational simplicity, write (a,b)+(c,d)=(a+c,b+d) As long as it is understood that the additions in G1 and G2 may not be the same binary operations.arrow_forwardExercises 12. Prove that the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers. (Hint: Consider the mapping defined by for all .)arrow_forward
- 13. Let be an abelian group with respect to multiplication. Prove that each of the following subsets of is subgroup of. a. b. for a fixed positive integer .arrow_forward32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping defined by is an automorphism of . Each of these automorphism is called an inner automorphism of . Prove that the set forms a normal subgroup of the group of all automorphism of . Exercise 20 of Section 3.5 20. For each in the group , define a mapping by . Prove that is an automorphism of .arrow_forwardFor each a in the group G, define a mapping ta:GG by ta(x)=axa1. Prove that ta is an automorphism of G. Sec. 4.6,32 Let a be a fixed element of the group G. According to Exercise 20 of Section 3.5, the mapping ta:GG defined by ta(x)=axa1 is an automorphism of G. Each of these automorphisms ta is called an inner automorphism of G. Prove that the set Inn(G)=taaG forms a normal subgroup of the group of all automorphisms of G.arrow_forward
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