Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 10, Problem 52E
Show that a homomorphism defined on a cyclic group is completelydetermined by its action on a generator of the group.
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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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- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.arrow_forwardLet 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_forward
- Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.arrow_forwardFind all homomorphic images of the quaternion group.arrow_forwardProve that if r and s are relatively prime positive integers, then any cyclic group of order rs is the direct sum of a cyclic group of order r and a cyclic group of order s.arrow_forward
- Prove 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_forward4. Prove that the special linear group is a normal subgroup of the general linear group .arrow_forwardIf G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.arrow_forward
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