Question 4.Given the triple < Z₂ x Z₁, B, D defined in terms ofthe Cartesian product Z₂ Z5 of two sets of congruence classes, Z₂ and Zs, under operations([k], [m])=([1], [n]) = ([k+l], [m+n])and([k], [m]) ([1], [n]) = ([kl], [mn])(a) Prove that the first distributive law holds true.(b) Hence prove that is a ring.(c) Is it a commutative ring? Justify your answer.

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Answered: Question 4. the Cartesian product Z₂ Z5…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 5E

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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 ]
31. Prove statement of Theorem : for all integers and .
Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.
In 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).
6. In Example 3 of section 3.1, find elements and of such that but . From Example 3 of section 3.1: and is a set of bijective functions defined on .
In Example 3 of Section 3.1, find all elements a of S(A) such that a2=e. From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of bijective functions defined on A.
For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H in Z18, partition Z18 into left cosets of H, and state the index [ Z18:H ] of H in Z18. H= [ 8 ] .
The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.
For each of the following values of n, find all distinct generators of the group Un described in Exercise 11. a. n=7 b. n=5 c. n=11 d. n=13 e. n=17 f. n=19

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