Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Chapter 10, Problem 15E
To determine
To calculate : All the elements that map to
Ker
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Chapter 10 Solutions
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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- Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an isomorphism from G to G that is different from the one given in Example 5 of this section. Example 5 Consider G=1,i,1,i under multiplication and G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. In order to define a mapping :G4 that is an isomorphism, one requirement is that must map the identity element 1 of G to the identity element [ 0 ] of 4 (part a of Theorem 3.30). Thus (1)=[ 0 ]. Another requirement is that inverses must map onto inverses (part b of Theorem 3.30). That is, if we take (i)=[ 1 ] then (i1)=((i))1=[ 1 ] Or (i)=[ 3 ] The remaining elements 1 in G and [ 2 ] in 4 are their own inverses, so we take (1)=[ 2 ]. Thus the mapping :G4 defined by (1)=[ 0 ], (i)=[ 1 ], (1)=[ 2 ], (i)=[ 3 ]arrow_forwardWrite 20 as the direct sum of two of its nontrivial subgroups.arrow_forwardExercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .arrow_forward
- 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.arrow_forward23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.arrow_forwardTrue or False Label each of the following statements as either true or false. Every isomorphism is a homomorphism.arrow_forward
- Let be as described in the proof of Theorem. Give a specific example of a positive element of .arrow_forwardLabel each of the following statements as either true or false. Every endomorphism is an epimorphism.arrow_forward7. Prove that on a given set of rings, the relation of being isomorphic has the reflexive, symmetric, and transitive properties.arrow_forward
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