Please send me aspsa <10:40Keshav NJIT on 12/22/22, 10:40 AMQuestion 12Graph Isomorphism:Write an Isomorphism s from G1 to G2,Write your answer as a list, comma separated, no spaces:s(1), s(2), s(3), s(4), s(5), s(6), s(7)G1:.5GºиG2:All MediaEP

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Answered: Question 12 Graph Isomorphism: Write an…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.8: Some Results On Finite Abelian Groups (optional)

Problem 2E

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29. Suppose , , represents a partition of the nonempty set A. Define R on A by if and only if there is a subset such that . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets .
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 ]
Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .
13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.
23. 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.
Prove that if f is a permutation on A, then (f1)1=f.
17. Find two groups and such that is a homomorphic image of but is not a homomorphic image of . (Thus the relation in Exercise does not have the symmetric property.) Exercise 15: 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.
42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .
6. Prove that if is a permutation on , then is a permutation on .
15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.
Label each of the following statements as either true or false. Every endomorphism is an epimorphism.

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Question 12 Graph Isomorphism: Write an Isomorphism s from G1 to G2, Write your answer as a list, comma separated, no spaces: s(1), s(2), s(3), s(4), s(5), s(6), s(7) G1: G2: