20. Let G be a group with subgroup H. Define a relationon G asfollows: a b if b-¹a E H. Prove that this is an equivalence relation(that is, reflexive, symmetric, and transitive). Prove that a~ b if andonly if aH=bH, and so the equivalence classes of this relation arethe cosets in G/H.

Answered: Image /qna-images/answer/7cbd0254-7916-470e-8f92-c80ce1084cbd.jpg

Answered: 2 20. Let G be a group with subgroup H.…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 44E: 44. Let be a subgroup of a group .For, define the relation by ...

See similar textbooks

Similar questions

Find two groups of order 6 that are not isomorphic.
Exercises 35. Prove that any two groups of order are isomorphic.
Let G be a group. Prove that the relation R on G, defined by xRy if and only if there exist an aG such that y=a1xa, is an equivalence relation. Let xG. Find [ x ], the equivalence class containing x, if G is abelian. (Sec 3.3,23) Sec. 3.3, #23: 23. Let R be the equivalence relation on G defined by xRy if and only if there exists an element a in G such that y=a1xa. If x(G), find [ x ], the equivalence class containing 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.
15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.
9. Suppose that and are subgroups of the abelian group such that . Prove that .
Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)
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.
44. Let be a subgroup of a group .For, define the relation by if and only if . Prove that is an equivalence relation on . Let . Find , the equivalence class containing .
Suppose that G and G are abelian groups such that G=H1H2 and G=H1H2. If H1 is isomorphic to H1 and H2 is isomorphic to H2, prove that G is isomorphic to G.
Label each of the following statements as either true or false. The relation of being a homomorphic image is an equivalence relation on a collection of groups.
24. Prove or disprove that every group of order is abelian.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

2 20. Let G be a group with subgroup H. Define a relation on G as follows: a b if b-¹a EH. Prove that this is an equivalence relation (that is, reflexive, symmetric, and transitive). Prove that a ~ b if and only if aH bH, and so the equivalence classes of this relation are the cosets in G/H. =