Qa Let 0:G-7 H be a homomorphism ofgoonent of finitegroups.that 1O4 divides 191.(6)-24 o isisaninice order paarepooveisomorphirms preve.anthat 101=191.elements gtG areandsuch th at 10(91=191 fadCव ए (जAomomodphistn 0?(C)-If allfinite.ofooderis'all geG,, whattheabout

It is given that, θ : G→H be a homomorphism. Then, we have to answer the following questions : a)…

Answered: Qo Let O: G-> H be a (a)- homomorphism…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 18E: Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the...

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Write 20 as the direct sum of two of its nontrivial subgroups.
11. Show that defined by is not a homomorphism.
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 ]
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.
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.
Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .
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.
A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
Prove that if f is a permutation on A, then (f1)1=f.
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 ].
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.
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).

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Qo Let O: G-> H be a (a)- homomorphism of an groumenta finite " order prove element that log) divides [g/. (b)- If O is an that 10(g) = 1g1. isomorphisms prove (c)_ If all elements gEG are finite order of is all gEG, what about the such that 10(g)] = [g) for can you say homomorphism 0?