The mapping: (Z6, +6) → (U4, ×4)ony vong ban ,0 is avouritaco atηTUOXsvong bps Its euouston aiIntify cadadtedSTONn→ 3" (mod 4)(G.e) sach that yois a homomorphism. (Here U4 = {1,3}.)(a) Write down the image of each element of Z6 under o.1 osT sula unel(b) Find Imo and Ker o.1401 = (512 jest, dous (II-) is aut(d) State a standard group that is isomorphic to the quotient group(c) Write down the cosets of Ker o in Z6-[8]ZG/Ker o.timil odi tadi orang of sluaham 8 - 058+

Answered: The mapping : (Z6, +6) → (U4, X4) on my…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 4E: Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an...

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Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
11. Show that defined by is not a homomorphism.
9. For any let denote in and let denote in . a. Prove that the mapping defined by is a homomorphism. b. Find ker .
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 ]
Let be as described in the proof of Theorem. Give a specific example of a positive element of .
Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
12. Consider the mapping defined by . Decide whether is a homomorphism, and justify your decision.
1. Consider , the groups of units in under multiplication. For each of the following subgroups in , partition into left cosets of , and state the index of in a. b.
Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.
28. For each, define by for. a. Show that is an element of . b. Let .Prove that is a subgroup of under mapping composition. c. Prove that is abelian, even though is not.
Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]
Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.

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