Question 9(a) Prove that every element of Q/Z has finite order.(b) Given two groups (G,) and (H, *). Suppose that is a homomorphism of G onto H. ForB◄ H and A := {g € G : 0(g) € B}, prove that A ◄ G.

This question is related to group theory.

Answered: (a) Prove that every element of Q/Z has…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 7TFE: Label each of the following statements as either true or false. Two groups can be isomorphic even...

See similar textbooks

Similar questions

Find two groups of order 6 that are not isomorphic.
Prove part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.
9. Suppose that and are subgroups of the abelian group such that . Prove that .
Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
1.Prove part of Theorem . Theorem 3.4: Properties of Group Elements Let be a group with respect to a binary operation that is written as multiplication. The identity element in is unique. For each, the inverse in is unique. For each . Reverse order law: For any and in ,. Cancellation laws: If and are in , then either of the equations or implies that .
32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping defined by is an automorphism of . Each of these automorphism is called an inner automorphism of . Prove that the set forms a normal subgroup of the group of all automorphism of . Exercise 20 of Section 3.5 20. For each in the group , define a mapping by . Prove that is an automorphism of .
Suppose that G and H are isomorphic groups. Prove that G is abelian if and only if H is abelian.
27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.
Exercises 22. Let be a finite cyclic group of order with generators and . Prove that the mapping is an automorphism of .
Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
Label each of the following statements as either true or false. Two groups can be isomorphic even though their group operations are different.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

(a) Prove that every element of Q/Z has finite order. (b) Given two groups (G, .) and (H, *). Suppose that is a homomorphism of G onto H. For BH and A := {g G: 0(g) E B}, prove that A◄G.